choco documentation

choco documentation
Choco solver
http://choco.mines-nantes.fr/
Documentation
^
^
O..
^
August 2, 2012
Copyright (C) 1999, 2011 F. Laburthe, N. Jussien.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3
or any later version published by the Free Software Foundation;
with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
A copy of the license is included in the section entitled ”GNU
Free Documentation License”.
CHOCO solver documentation
BSD licence 2012
-ii-
2/8/2012
CONTENTS
Contents
Preface
1
I
3
Documentation
1 Introduction to constraint programming and Choco
1.1 About constraint programming . . . . . . . . . . . . . . . .
1.1.1 Constraints . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Constraint Programming . . . . . . . . . . . . . . .
1.2 Modeling with Constraint programming . . . . . . . . . . .
1.2.1 The Constraint Satisfaction Problem . . . . . . . . .
1.2.2 Examples of CSP models . . . . . . . . . . . . . . .
1.3 My first Choco program: the magic square . . . . . . . . . .
1.3.1 The magic square problem . . . . . . . . . . . . . . .
1.3.2 A mathematical model . . . . . . . . . . . . . . . . .
1.3.3 To Choco... . . . . . . . . . . . . . . . . . . . . . . .
1.3.4 The program . . . . . . . . . . . . . . . . . . . . . .
1.3.5 In summary . . . . . . . . . . . . . . . . . . . . . . .
1.4 Complete examples . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Example 1: the n-queens problem with Choco . . . .
1.4.2 Example 2: the ternary Steiner problem with Choco
1.4.3 Example 3: the CycloHexane problem with Choco .
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2 The model
2.1 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Simple Variables . . . . . . . . . . . . . . . . . . . . .
2.1.2 Constants . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Expression variables and operators . . . . . . . . . . .
2.1.4 MultipleVariables . . . . . . . . . . . . . . . . . . . . .
2.1.5 Decision/non-decision variables . . . . . . . . . . . . .
2.1.6 Objective variable . . . . . . . . . . . . . . . . . . . .
2.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Binary constraints . . . . . . . . . . . . . . . . . . . .
2.2.2 Ternary constraints . . . . . . . . . . . . . . . . . . . .
2.2.3 Constraints involving real variables . . . . . . . . . . .
2.2.4 Constraints involving set variables . . . . . . . . . . .
2.2.5 Channeling constraints . . . . . . . . . . . . . . . . . .
2.2.6 Constraints in extension and relations . . . . . . . . .
2.2.7 Reified constraints . . . . . . . . . . . . . . . . . . . .
2.2.8 Global constraints . . . . . . . . . . . . . . . . . . . .
2.2.9 Things to know about Model, Variable and Constraint
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CHOCO solver documentation
BSD licence 2012
-iii/216-
2/8/2012
CONTENTS
3 The solver
3.1 Variables reading . . . . . . . . . . . . . . . . . . . . . .
3.1.1 from IntegerVariable toIntDomainVar . . . . .
3.1.2 from SetVariable to SetVar . . . . . . . . . . .
3.1.3 from RealVariable to RealVar . . . . . . . . . .
3.1.4 from TaskVariable to TaskVar . . . . . . . . . .
3.2 Constraints reading . . . . . . . . . . . . . . . . . . . . .
3.3 Solve a problem . . . . . . . . . . . . . . . . . . . . . . .
3.4 Storing et restoring solutions . . . . . . . . . . . . . . .
3.5 Search Strategy . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Overriding the default search strategy . . . . . .
3.5.2 Pre-defined search strategies . . . . . . . . . . .
3.5.3 Why is it important to define a search strategy ?
3.5.4 Restarts . . . . . . . . . . . . . . . . . . . . . . .
3.6 Limiting Search Space . . . . . . . . . . . . . . . . . . .
3.7 Logging the search . . . . . . . . . . . . . . . . . . . . .
3.7.1 Architecture . . . . . . . . . . . . . . . . . . . .
3.7.2 Verbosities and messages . . . . . . . . . . . . .
3.7.3 Basic settings . . . . . . . . . . . . . . . . . . . .
3.8 Clean a Solver . . . . . . . . . . . . . . . . . . . . . . .
3.8.1 What about simply calling solver.clear()? . . .
3.8.2 Things to know about Solver reusability . . . . .
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4 Advanced uses of Choco
4.1 Environment . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Copying . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Trailing . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Backtrackable structures . . . . . . . . . . . . . . .
4.2 How does the propagation engine work ? . . . . . . . . . .
4.2.1 How does a search loop work ? . . . . . . . . . . .
4.2.2 Propagate . . . . . . . . . . . . . . . . . . . . . . .
4.3 Define your own search strategy . . . . . . . . . . . . . . .
4.3.1 How to define your own Branching object . . . . .
4.3.2 Define your own variable selection . . . . . . . . .
4.3.3 Define your own value selection . . . . . . . . . . .
4.4 Define your own limit search space . . . . . . . . . . . . .
4.5 Define your own constraint . . . . . . . . . . . . . . . . .
4.5.1 The constraint hierarchy . . . . . . . . . . . . . . .
4.5.2 Example: implement and add the IsOdd constraint
4.5.3 Example of an empty constraint . . . . . . . . . .
4.6 Define your own operator . . . . . . . . . . . . . . . . . .
4.7 Define your own variable . . . . . . . . . . . . . . . . . . .
4.8 Model and Solver detectors . . . . . . . . . . . . . . . . .
4.8.1 Model detector . . . . . . . . . . . . . . . . . . . .
4.8.2 Preprocess solver . . . . . . . . . . . . . . . . . . .
4.9 Logging statements . . . . . . . . . . . . . . . . . . . . . .
4.9.1 Define your own logger. . . . . . . . . . . . . . . .
4.9.2 How to write logging statements ? . . . . . . . . .
4.9.3 Handlers. . . . . . . . . . . . . . . . . . . . . . . .
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5 Choco and CP-Viz
5.1 Three step tutorial . . . . . .
5.2 Visualization and Visualizers
5.2.1 Visualization . . . . .
5.2.2 Visualizers . . . . . .
5.3 Example of code integration .
CHOCO solver documentation
BSD licence 2012
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2/8/2012
CONTENTS
II
Elements of Choco
6 Variables (Model)
6.1 Integer variables
6.2 Real variables . .
6.3 Set variables . .
6.4 Task variables . .
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7 Operators (Model)
7.1 abs (operator) . . . . .
7.2 cos (operator) . . . . .
7.3 div (operator) . . . . .
7.4 ifThenElse (operator)
7.5 max (operator) . . . .
7.6 min (operator) . . . .
7.7 minus (operator) . . .
7.8 mod (operator) . . . .
7.9 mult (operator) . . . .
7.10 neg (operator) . . . .
7.11 plus (operator) . . . .
7.12 power (operator) . . .
7.13 scalar (operator) . . .
7.14 sin (operator) . . . . .
7.15 sum (operator) . . . .
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81
81
81
82
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83
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84
85
85
86
87
87
87
8 Constraints (Model)
8.1 abs (constraint) . . . . . . . . .
8.2 allDifferent (constraint) . . . .
8.3 among (constraint) . . . . . . .
8.4 and (constraint) . . . . . . . .
8.5 atMostNValue (constraint) . .
8.6 boolChanneling (constraint) . .
8.7 clause (constraint) . . . . . . .
8.8 costRegular (constraint) . . . .
8.9 cumulative (constraint) . . . .
8.10 cumulativeMax (constraint) . .
8.11 cumulativeMin (constraint) . .
8.12 disjoint (constraint) . . . . . .
8.13 disjunctive (constraint) . . . . .
8.14 distanceEQ (constraint) . . . .
8.15 distanceGT (constraint) . . . .
8.16 distanceLT (constraint) . . . .
8.17 distanceNEQ (constraint) . . .
8.18 domainChanneling (constraint)
8.19 element (constraint) . . . . . .
8.20 endsAfter (constraint) . . . . .
8.21 endsAfterBegin (constraint) . .
8.22 endsAfterEnd (constraint) . . .
8.23 endsBefore (constraint) . . . .
8.24 endsBeforeBegin (constraint) .
8.25 endsBeforeEnd (constraint) . .
8.26 endsBetween (constraint) . . .
8.27 eq (constraint) . . . . . . . . .
8.28 eqCard (constraint) . . . . . . .
8.29 equation (constraint) . . . . . .
8.30 exactly (constraint) . . . . . . .
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89
89
89
90
91
92
93
93
94
97
98
99
100
100
101
102
102
103
104
104
104
105
105
105
106
106
106
107
108
108
109
CHOCO solver documentation
BSD licence 2012
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-v-
2/8/2012
CONTENTS
8.31
8.32
8.33
8.34
8.35
8.36
8.37
8.38
8.39
8.40
8.41
8.42
8.43
8.44
8.45
8.46
8.47
8.48
8.49
8.50
8.51
8.52
8.53
8.54
8.55
8.56
8.57
8.58
8.59
8.60
8.61
8.62
8.63
8.64
8.65
8.66
8.67
8.68
8.69
8.70
8.71
8.72
8.73
8.74
8.75
8.76
8.77
8.78
8.79
8.80
8.81
8.82
FALSE (constraint) . . . . . . . . . . . . . .
feasPairAC (constraint) . . . . . . . . . . .
feasTupleAC (constraint) . . . . . . . . . .
feasTupleFC (constraint) . . . . . . . . . . .
forbiddenInterval (constraint) . . . . . . . .
geost (constraint) . . . . . . . . . . . . . . .
geq (constraint) . . . . . . . . . . . . . . . .
geqCard (constraint) . . . . . . . . . . . . .
globalCardinality (constraint) . . . . . . . .
gt (constraint) . . . . . . . . . . . . . . . .
ifOnlyIf (constraint) . . . . . . . . . . . . .
ifThenElse (constraint) . . . . . . . . . . . .
implies (constraint) . . . . . . . . . . . . . .
increasingNValue (constraint) . . . . . . . .
increasingSum (constraint) . . . . . . . . .
infeasPairAC (constraint) . . . . . . . . . .
infeasTupleAC (constraint) . . . . . . . . .
infeasTupleFC (constraint) . . . . . . . . .
intDiv (constraint) . . . . . . . . . . . . . .
inverseChanneling (constraint) . . . . . . .
inverseChannelingWithinRange (constraint)
inverseSet (constraint) . . . . . . . . . . . .
isIncluded (constraint) . . . . . . . . . . . .
isNotIncluded (constraint) . . . . . . . . . .
knapsackProblem (constraint) . . . . . . . .
leq (constraint) . . . . . . . . . . . . . . . .
leqCard (constraint) . . . . . . . . . . . . .
lex (constraint) . . . . . . . . . . . . . . . .
lexChain (constraint) . . . . . . . . . . . . .
lexChainEq (constraint) . . . . . . . . . . .
lexEq (constraint) . . . . . . . . . . . . . .
leximin (constraint) . . . . . . . . . . . . .
lt (constraint) . . . . . . . . . . . . . . . . .
max (constraint) . . . . . . . . . . . . . . .
8.64.1 max of a list . . . . . . . . . . . . .
8.64.2 max of a set . . . . . . . . . . . . . .
member (constraint) . . . . . . . . . . . . .
min (constraint) . . . . . . . . . . . . . . .
8.66.1 min of a list . . . . . . . . . . . . . .
8.66.2 min of a set . . . . . . . . . . . . . .
mod (constraint) . . . . . . . . . . . . . . .
multiCostRegular (constraint) . . . . . . . .
nand (constraint) . . . . . . . . . . . . . . .
neq (constraint) . . . . . . . . . . . . . . . .
neqCard (constraint) . . . . . . . . . . . . .
not (constraint) . . . . . . . . . . . . . . . .
notMember (constraint) . . . . . . . . . . .
nor (constraint) . . . . . . . . . . . . . . . .
nth (constraint) . . . . . . . . . . . . . . . .
occurrence (constraint) . . . . . . . . . . . .
occurrenceMax (constraint) . . . . . . . . .
occurrenceMin (constraint) . . . . . . . . .
oppositeSign (constraint) . . . . . . . . . .
or (constraint) . . . . . . . . . . . . . . . .
pack (constraint) . . . . . . . . . . . . . . .
precedenceReified (constraint) . . . . . . . .
CHOCO solver documentation
BSD licence 2012
-vi-
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109
109
110
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148
2/8/2012
CONTENTS
8.83 precedenceimplied (constraint)
8.84 precedence (constraint) . . . .
8.85 precedencedisjoint (constraint)
8.86 regular (constraint) . . . . . . .
8.87 reifiedAnd (constraint) . . . . .
8.88 reifiedConstraint (constraint) .
8.89 reifiedLeftImp (constraint) . . .
8.90 reifiedNot (constraint) . . . . .
8.91 reifiedOr (constraint) . . . . . .
8.92 reifiedRightImp (constraint) . .
8.93 reifiedXnor (constraint) . . . .
8.94 reifiedXor (constraint) . . . . .
8.95 relationPairAC (constraint) . .
8.96 relationTupleAC (constraint) .
8.97 relationTupleFC (constraint) .
8.98 sameSign (constraint) . . . . .
8.99 setDisjoint (constraint) . . . . .
8.100setInter (constraint) . . . . . .
8.101setUnion (constraint) . . . . . .
8.102sorting (constraint) . . . . . . .
8.103startsAfter (constraint) . . . .
8.104startsAfterBegin (constraint) .
8.105startsAfterEnd (constraint) . .
8.106startsBefore (constraint) . . . .
8.107startsBeforeBegin (constraint) .
8.108startsBeforeEnd (constraint) . .
8.109startsBetween (constraint) . . .
8.110stretchPath (constraint) . . . .
8.111times (constraint) . . . . . . . .
8.112tree (constraint) . . . . . . . .
8.113TRUE (constraint) . . . . . . .
8.114xnor (constraint) . . . . . . . .
8.115xor (constraint) . . . . . . . . .
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9 Options (Model)
9.1 Options and settings . . . . . . . . . . . . .
9.1.1 Options for variables: . . . . . . . .
9.1.2 Options for expressions: . . . . . . .
9.1.3 Options and settings for constraints:
9.1.4 Options for solvers: . . . . . . . . . .
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10 Branching strategies (Solver)
10.1 AssignInterval (Branching strategy) . . . . . . . . . . .
10.2 AssignOrForbidIntVarVal (Branching strategy) . . . . .
10.3 AssignOrForbidIntVarValPair (Branching strategy) . . .
10.4 AssignSetVar (Branching strategy) . . . . . . . . . . . .
10.5 AssignVar (Branching strategy) . . . . . . . . . . . . . .
10.6 DomOverWDegBranchingNew (Branching strategy) . .
10.7 DomOverWDegBinBranchingNew (Branching strategy)
10.8 ImpactBasedBranching (Branching strategy) . . . . . .
10.9 PackDynRemovals (Branching strategy) . . . . . . . . .
10.10SetTimes (Branching strategy) . . . . . . . . . . . . . .
10.11TaskOverWDegBinBranching (Branching strategy) . . .
CHOCO solver documentation
BSD licence 2012
-vii-
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2/8/2012
CONTENTS
11 Variable selectors (Solver)
11.1 CompositeIntVarSelector (Variable selector)
11.2 CyclicRealVarSelector (Variable selector) .
11.3 LexIntVarSelector (Variable selector) . . . .
11.4 MaxDomain (Variable selector) . . . . . . .
11.5 MaxDomSet (Variable selector) . . . . . . .
11.6 MaxRegret (Variable selector) . . . . . . . .
11.7 MaxRegretSet (Variable selector) . . . . . .
11.8 MaxValueDomain (Variable selector) . . . .
11.9 MaxValueDomSet (Variable selector) . . . .
11.10MinDomain (Variable selector) . . . . . . .
11.11MinDomSet (Variable selector) . . . . . . .
11.12MinValueDomain (Variable selector) . . . .
11.13MinValueDomSet (Variable selector) . . . .
11.14MostConstrained (Variable selector) . . . .
11.15MostConstrainedSet (Variable selector) . .
11.16RandomIntVarSelector (Variable selector) .
11.17RandomSetVarSelector (Variable selector) .
11.18StaticSetVarOrder (Variable selector) . . . .
11.19StaticVarOrder (Variable selector) . . . . .
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12 Value iterators (Solver)
189
12.1 DecreasingDomain (Value iterator) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
12.2 IncreasingDomain (Value iterator) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
12.3 RealIncreasingDomain (Value iterator) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
13 Value selector (Solver)
13.1 BestFit (Value selector) . . . . . . . . . .
13.2 CostRegularValSelector (Value selector) .
13.3 FCostRegularValSelector (Value selector)
13.4 MaxVal (Value selector) . . . . . . . . . .
13.5 MidVal (Value selector) . . . . . . . . . .
13.6 MinEnv (Value selector) . . . . . . . . . .
13.7 MinVal (Value selector) . . . . . . . . . .
13.8 RandomIntValSelector (Value selector) . .
13.9 RandomSetValSelector (Value selector) . .
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191
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14 Visualizer (Visualization)
14.1 Variable-oriented Visualizers . . . . . .
14.1.1 BinaryMatrix (visualizer) . . .
14.1.2 BinaryVector (visualizer) . . .
14.1.3 DomainMatrix (visualizer) . . .
14.1.4 Vector (visualizer) . . . . . . .
14.1.5 VectorSize (visualizer) . . . . .
14.1.6 VectorWaterfall (visualizer) . .
14.2 Constraint-oriented Visualizers . . . .
14.2.1 AllDifferent (visualizer) . . . .
14.2.2 AllDifferentMatrix (visualizer)
14.2.3 BooleanChanneling (visualizer)
14.2.4 Cumulative (visualizer) . . . .
14.2.5 Element (visualizer) . . . . . .
14.2.6 Gcc (visualizer) . . . . . . . . .
14.3 Inverse (visualizer) . . . . . . . . . . .
14.3.1 LexLe (visualizer) . . . . . . .
14.3.2 LexLt (visualizer) . . . . . . .
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CHOCO solver documentation
BSD licence 2012
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2/8/2012
CONTENTS
III
Extras
201
15 Sudoku and Constraint Programming
15.1 Sudoku ?!? . . . . . . . . . . . . . . . . . . . .
15.1.1 Solving sudokus . . . . . . . . . . . . .
15.2 Sudoku and Artificial Intelligence . . . . . . . .
15.2.1 Simple rules: single candidate and single
15.2.2 Human reasoning principles . . . . . . .
15.2.3 Towards Constraint Programming . . .
15.3 See also . . . . . . . . . . . . . . . . . . . . . .
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Glossary
207
Bibliography
207
GNU Free Documentation License
1. APPLICABILITY AND DEFINITIONS . . . . . . . . .
2. VERBATIM COPYING . . . . . . . . . . . . . . . . . .
3. COPYING IN QUANTITY . . . . . . . . . . . . . . . .
4. MODIFICATIONS . . . . . . . . . . . . . . . . . . . . .
5. COMBINING DOCUMENTS . . . . . . . . . . . . . . .
6. COLLECTIONS OF DOCUMENTS . . . . . . . . . . .
7. AGGREGATION WITH INDEPENDENT WORKS . .
8. TRANSLATION . . . . . . . . . . . . . . . . . . . . . .
9. TERMINATION . . . . . . . . . . . . . . . . . . . . . .
10. FUTURE REVISIONS OF THIS LICENSE . . . . . .
11. RELICENSING . . . . . . . . . . . . . . . . . . . . . .
ADDENDUM: How to use this License for your documents
CHOCO solver documentation
BSD licence 2012
-ix-
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211
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216
2/8/2012
CONTENTS
Preface
Choco is a java library for constraint satisfaction problems (CSP) and constraint programming (CP). It
is built on a event-based propagation mechanism with backtrackable structures. Choco is an open-source
software, distributed under a BSD licence and hosted by sourceforge.net. For any informations visit
http://choco.emn.fr.
This document is organized as follows:
• Documentation is the user-guide of Choco. After a short introduction to constraint programming
and to the Choco solver, it presents the basics of modeling and solving with Choco, and some
advanced usages (customizing propagation and search).
• Elements of Choco gives a detailed description of the variables, operators, constraints currently
available in Choco.
• Extras presents future works, only available on the beta version or extension of the current jar,
such as the visualization module of Choco. The section dedicated to Sudoku aims at explaining
the basic principles of Constraint Programming (propagation and search) on this famous game.
CHOCO solver documentation
BSD licence 2012
-1/216-
2/8/2012
Part I
Documentation
CHOCO solver documentation
BSD licence 2012
-3/216-
2/8/2012
The documentation of Choco is organized as follows:
• The concise introduction provides some informations about constraint programming concepts and
a “Hello world”-like first Choco program.
• The model section gives informations on how to create a model and introduces variables and
constraints.
• The solver section gives informations on how to create a solver, to read a model, to define a search
strategy, and finally to solve a problem.
• The advanced use section explains how to define your own limit search space, search strategy,
constraint, operator, variable, backtrackable structure and write logging statements.
CHOCO solver documentation
BSD licence 2012
-5/216-
2/8/2012
Chapter 1
Introduction to constraint
programming and Choco
1.1
About constraint programming
Constraint programming represents one of the closest approaches computer science has yet made
to the Holy Grail of programming: the user states the problem, the computer solves it.
E. C. Freuder, Constraints, 1997.
Fast increasing computing power in the 1960s led to a wealth of works around problem solving,
at the root of Operational Research, Numerical Analysis, Symbolic Computing, Scientific Computing,
and a large part of Artificial Intelligence and programming languages. Constraint Programming is a
discipline that gathers, interbreeds, and unifies ideas shared by all these domains to tackle decision
support problems.
Constraint programming has been successfully applied in numerous domains. Recent applications
include computer graphics (to express geometric coherence in the case of scene analysis), natural language
processing (construction of efficient parsers), database systems (to ensure and/or restore consistency
of the data), operations research problems (scheduling, routing), molecular biology (DNA sequencing),
business applications (option trading), electrical engineering (to locate faults), circuit design (to compute
layouts), etc.
Current research in this area deals with various fundamental issues, with implementation aspects and
with new applications of constraint programming.
1.1.1
Constraints
A constraint is simply a logical relation among several unknowns (or variables), each taking a value in a
given domain. A constraint thus restricts the possible values that variables can take, it represents some
partial information about the variables of interest. For instance, the circle is inside the square relates
two objects without precisely specifying their positions, i.e., their coordinates. Now, one may move the
square or the circle and one is still able to maintain the relation between these two objects. Also, one
may want to add another object, say a triangle, and to introduce another constraint, say the square is to
the left of the triangle. From the user (human) point of view, everything remains absolutely transparent.
Constraints naturally meet several interesting properties:
• constraints may specify partial information, i.e. constraint need not uniquely specify the values of
its variables,
• constraints are non-directional, typically a constraint on (say) two variables X, Y can be used to
infer a constraint on X given a constraint on Y and vice versa,
CHOCO solver documentation
BSD licence 2012
-7/216-
2/8/2012
CHAPTER 1. INTRODUCTION TO CONSTRAINT PROGRAMMING AND CHOCO
• constraints are declarative, i.e. they specify what relationship must hold without specifying a
computational procedure to enforce that relationship,
• constraints are additive, i.e. the order of imposition of constraints does not matter, all that matters
at the end is that the conjunction of constraints is in effect,
• constraints are rarely independent, typically constraints in the constraint store share variables.
Constraints arise naturally in most areas of human endeavor. The three angles of a triangle sum to
180 degrees, the sum of the currents flowing into a node must equal zero, the position of the scroller in
the window scrollbar must reflect the visible part of the underlying document, these are some examples
of constraints which appear in the real world. Thus, constraints are a natural medium for people to
express problems in many fields.
1.1.2
Constraint Programming
Constraint programming is the study of computational systems based on constraints. The idea of constraint programming is to solve problems by stating constraints (conditions, properties) which must be
satisfied by the solution.
Work in this area can be tracked back to research in Artificial Intelligence and Computer Graphics in
the sixties and seventies. Only in the last decade, however, has there emerged a growing realization that
these ideas provide the basis for a powerful approach to programming, modeling and problem solving
and that different efforts to exploit these ideas can be unified under a common conceptual and practical
framework, constraint programming.
If you know sudoku, then you know constraint programming. See
why here.
1.2
Modeling with Constraint programming
The formulation and the resolution of combinatorial problems are the two main goals of the constraint
programming domain. This is an essential way to solve many interesting industrial problems such as
scheduling, planning or design of timetables.
1.2.1
The Constraint Satisfaction Problem
Constraint programming allows to solve combinatorial problems modeled by a Constraint Satisfaction
Problem (CSP). Formally, a CSP is defined by a triplet (X, D, C):
• Variables: X = {X1 , X2 , . . . , Xn } is the set of variables of the problem.
• Domains: D is a function which associates to each variable Xi its domain D(Xi ), i.e. the set
of possible values that can be assigned to Xi . The domain of a variable is usually a finite set of
integers: D(Xi ) ⊂ Z (integer variable). But a domain can also be continuous (D(Xi ) ⊆ R for a
real variable) or made of discrete set values (D(Xi ) ⊆ P(Z) for a set variable).
• Constraints: C = {C1 , C2 , . . . , Cm } is the set of constraints. A constraint Cj is a relation defined
on a subset X j = {X1j , X2j , . . . , Xnj j } ⊆ X of variables which restricts the possible tuples of values
(v1 , . . . , vnj ) for these variables:
(v1 , . . . , vnj ) ∈ Cj ∩ (D(X1j ) × D(X2j ) × · · · × D(Xnj j )).
Such a relation can be defined explicitely (ex: (X1 , X2 ) ∈ {(0, 1), (1, 0)}) or implicitely (ex: X1 +
X2 ≤ 1).
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Solving a CSP consists in finding a tuple v = (v1 , . . . , vn ) ∈ D(X) on the set of variables which
satisfies all the constraints:
(v1 , . . . , vnj ) ∈ Cj ,
∀j ∈ {1, . . . , m}.
For optimization problems, one needs to define an objective function f : D(X) → R. An optimal
solution is then a solution tuple of the CSP that minimizes (or maximizes) function f .
1.2.2
Examples of CSP models
This part provides three examples using different types of variables in different problems. These examples
are used throughout this tutorial to illustrate their modeling with Choco.
Example 1: the n-queens problem.
Let us consider a chess board with n rows and n columns. A queen can move as far as she pleases,
horizontally, vertically, or diagonally. The standard n-queens problem asks how to place n queens on an
n-ary chess board so that none of them can hit any other in one move.
The n-queens problem can be modeled by a CSP in the following way:
• Variables: X = {Xi | i ∈ [1, n]}; one variable represents a column and the constraint ”queens
must be on different columns” is induced by the variables.
• Domain: for all variable Xi ∈ X, D(Xi ) = {1, 2, . . . , n}.
• Constraints: the set of constraints is defined by the union of the three following constraints,
– queens have to be on distinct lines:
∗ Clines = {Xi 6= Xj | i, j ∈ [1, n], i 6= j}.
– queens have to be on distinct diagonals:
∗ Cdiag1 = {Xi 6= Xj + j − i | i, j ∈ [1, n], i 6= j}.
∗ Cdiag2 = {Xi 6= Xj + i − j | i, j ∈ [1, n], i 6= j}.
Example 2: the ternary Steiner problem.
A ternary Steiner system of order n is a set of n ∗ (n − 1)/6 triplets of distinct elements taking their
values in [1, n], such that all the pairs included in two distinct triplets are different. See http://en.
wikipedia.org/wiki/Steiner_system for details.
The ternary Steiner problem can be modeled by a CSP in the following way:
• let t = n ∗ (n − 1)/6.
• Variables: X = {Xi | i ∈ [1, t]}.
• Domain: for all i ∈ [1, t], D(Xi ) = {1, ..., n}.
• Constraints:
– every set variable Xi has a cardinality of 3:
∗ for all i ∈ [1, t], |Xi | = 3.
– the cardinality of the intersection of every two distinct sets must not exceed 1:
∗ for all i, j ∈ [1, t], i 6= j, |Xi ∩ Xj | ≤ 1.
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Example 3: the CycloHexane problem.
The problem consists in finding the 3D configuration of a cyclohexane molecule. It is described with a
system of three non linear equations:
• Variables: x, y, z.
• Domain: ] − ∞; +∞[.
• Constraints:
y 2 ∗ (1 + z 2 ) + z ∗ (z − 24 ∗ y) = −13
x2 ∗ (1 + y 2 ) + y ∗ (y − 24 ∗ x) = −13
z 2 ∗ (1 + x2 ) + x ∗ (x − 24 ∗ z) = −13
1.3
1.3.1
My first Choco program: the magic square
The magic square problem
In the following, we will address the magic square problem of order 3 to illustrate step-by-step how to
model and solve this problem using choco.
Definition:
A magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such
that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A standard
magic square contains the integers from 1 to n2 .
The constant sum in every row, column and diagonal is called the magic constant or magic sum M .
The magic constant of a classic magic square depends only on n and has the value: M (n) = n(n2 + 1)/2.
More details on the magic square problem.
1.3.2
A mathematical model
Let xij be the variable indicating the value of the j th cell of row i. Let C be the set of constraints
modeling the magic square as:
xij ∈ [1, n2 ],
∀i, j ∈ [1, n]
xij 6= xkl ,
n
X
xij = n(n2 + 1)/2,
∀i, j, k, l ∈ [1, n], i 6= k, j 6= l
j=1
n
X
i=1
n
X
∀i ∈ [1, n]
xij = n(n2 + 1)/2,
∀j ∈ [1, n]
xii = n(n2 + 1)/2
i=1
1
X
xi(n−i) = n(n2 + 1)/2
i=n
We have all the required information to model the problem with Choco.
For the moment, we just talk about model translation from a mathematical representation to Choco. Choco can be used as a black box, that means
we just need to define the problem without knowing the way it will be
solved. We can therefore focus on the modeling not on the solving.
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1.3.3
To Choco...
First, we need to know some of the basic Choco objects:
• The model (object Model in Choco) is one of the central elements of a Choco program. Variables
and constraints are associated to it.
• The variables (objects IntegerVariable, SetVariable, and RealVariable in Choco) are the
unknowns of the problem. Values of variables are taken from a domain which is defined by a set
of values or quite often simply by a lower bound and an upper bound of the allowed values. The
domain is given when creating the variable.
Do not forget that we manipulate variables in the mathematical
sense (as opposed to classical computer science). Their effective value
will be known only once the problem has been solved.
• The constraints define relations to be satisfied between variables and constants. In our first
model, we only use the following constraints provided by Choco:
– eq(var1, var2) which ensures that var1 equals var2.
– neq(var1, var2) which ensures that var1 is not equal to var2.
– sum(var[]) which returns expression var[0]+var[1]+...+var[n].
1.3.4
The program
After having created your java class file, import the Choco class to use the API:
import choco.Choco;
First of all, let’s create a Model:
// Constant declaration
int n = 3; // Order of the magic square
int magicSum = n * (n * n + 1) / 2; // Magic sum
// Build the model
CPModel m = new CPModel();
We create an instance of CPModel() for Constraint Programming Model. Do not forget to add the
following imports:
import choco.cp.model.CPModel;
Then we declare the variables of the problem:
// Creation of an array of variables
IntegerVariable[][] var = new IntegerVariable[n][n];
// For each variable, we define its name and the boundaries of its domain.
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
var[i][j] = Choco.makeIntVar("var_" + i + "_" + j, 1, n * n);
// Associate the variable to the model.
m.addVariable(var[i][j]);
}
}
Add the import:
import choco.kernel.model.variables.integer.IntegerVariable;
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We have defined the variable using the makeIntVar method which creates an enumerated domain: all
the values are stored in the java object (beware, it is usually not necessary to store all the values and it
is less efficient than to create a bounded domain).
Now, we are going to state a constraint ensuring that all variables must have a different value:
// All cells of the matrix must be different
for (int i = 0; i < n * n; i++) {
for (int j = i + 1; j < n * n; j++) {
Constraint c = (Choco.neq(var[i / n][i % n], var[j / n][j % n]));
m.addConstraint(c);
}
}
Add the import:
import choco.kernel.model.constraints.Constraint;
Then, we add the constraint ensuring that the magic sum is respected:
// All row’s sum has to be equal to the magic sum
for (int i = 0; i < n; i++) {
m.addConstraint(Choco.eq(Choco.sum(var[i]), magicSum));
}
Then we define the constraint ensuring that each column is equal to the magic sum. Actually, var just
denotes the rows of the square. So we have to declare a temporary array of variables that defines the
columns.
IntegerVariable[][] varCol = new IntegerVariable[n][n];
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
// Copy of var in the column order
varCol[i][j] = var[j][i];
}
// All column’s sum is equal to the magic sum
m.addConstraint(Choco.eq(Choco.sum(varCol[i]), magicSum));
}
It is sometimes useful to define some temporary variables to keep the model simple or to reorder array
of variables. That is why we also define two other temporary arrays for diagonals.
IntegerVariable[] varDiag1 = new IntegerVariable[n];
IntegerVariable[] varDiag2 = new IntegerVariable[n];
for (int i = 0; i < n; i++) {
varDiag1[i] = var[i][i]; // Copy of var in varDiag1
varDiag2[i] = var[(n - 1) - i][i]; // Copy of var in varDiag2
}
// All diagonal’s sum has to be equal to the magic sum
m.addConstraint(Choco.eq(Choco.sum(varDiag1), magicSum));
m.addConstraint(Choco.eq(Choco.sum(varDiag2), magicSum));
Now, we have defined the model. The next step is to solve it. For that, we build a Solver object:
// Build the solver
CPSolver s = new CPSolver();
with the imports:
import choco.cp.solver.CPSolver;
We create an instance of CPSolver() for Constraint Programming Solver. Then, the solver reads (translates) the model and solves it:
// Read the model
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s.read(m);
// Solve the model
s.solve();
// Print the solution
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
System.out.print(MessageFormat.format("{0} ", s.getVar(var[i][j]).getVal()));
}
System.out.println();
}
The only variables that need to be printed are the ones in var (all the others are only references to these
ones).
We have to use the Solver to get the value of each variable of the model.
The Model only declares the objects, the Solver finds their value.
We are done, we have created our first Choco program.
1.3.5
In summary
• A Choco Model is defined by a set of Variables with a given domain and a set of Constraints that
link Variables: it is necessary to add both Variables and Constraints to the Model.
• temporary Variables are useful to keep the Model readable, or necessary when reordering arrays.
• The value of a Variable can be known only once the Solver has found a solution.
• To keep the code readable, you can avoid the calls to the static methods of the Choco classes, by
importing the static classes, i.e. instead of:
import choco.Choco;
...
IntegerVariable v = Choco.makeIntVar("v", 1, 10);
...
Constraint c = Choco.eq(v, 5);
you can use:
import static choco.Choco.*;
...
IntegerVariable v = makeIntVar("v", 1, 10);
...
Constraint c = eq(v, 5);
1.4
Complete examples
We provide now the complete Choco model for the three examples previously described.
1.4.1
Example 1: the n-queens problem with Choco
This first model for the n-queens problem only involves binary constraints of differences between integer
variables. One can immediately recognize the 4 main elements of any Choco code. First of all, create
the model object. Then create the variables by using the Choco API (One variable per queen giving the
row (or the column) where the queen will be placed). Finally, add the constraints and solve the problem.
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int nbQueen = 8;
//1- Create the model
CPModel m = new CPModel();
//2- Create the variables
IntegerVariable[] queens = Choco.makeIntVarArray("Q", nbQueen, 1, nbQueen);
//3- Post constraints
for (int i = 0; i < nbQueen; i++) {
for (int j = i + 1; j < nbQueen; j++) {
int k = j - i;
m.addConstraint(Choco.neq(queens[i], queens[j]));
m.addConstraint(Choco.neq(queens[i], Choco.plus(queens[j], k))); // diagonal
constraints
m.addConstraint(Choco.neq(queens[i], Choco.minus(queens[j], k))); // diagonal
constraints
}
}
//4- Create the solver
CPSolver s = new CPSolver();
s.read(m);
s.solveAll();
//5- Print the number of solutions found
System.out.println("Number of solutions found:"+s.getSolutionCount());
1.4.2
Example 2: the ternary Steiner problem with Choco
The ternary Steiner problem is entirely modeled using set variables and set constraints.
//1- Create the problem
CPModel mod = new CPModel();
int m = 7;
int n = m * (m - 1) / 6;
//2- Create Variables
SetVariable[] vars = new SetVariable[n]; // A variable for each set
SetVariable[] intersect = new SetVariable[n * n]; // A variable for each pair of sets
for (int i = 0; i < n; i++)
vars[i] = Choco.makeSetVar("set " + i, 1, n);
for (int i = 0; i < n; i++)
for (int j = i + 1; j < n; j++)
intersect[i * n + j] = Choco.makeSetVar("interSet " + i + " " + j, 1, n);
//3- Post constraints
for (int i = 0; i < n; i++)
mod.addConstraint(Choco.eqCard(vars[i], 3));
for (int i = 0; i < n; i++)
for (int j = i + 1; j < n; j++) {
// the cardinality of the intersection of each pair is equal to one
mod.addConstraint(Choco.setInter(vars[i], vars[j], intersect[i * n + j]));
mod.addConstraint(Choco.leqCard(intersect[i * n + j], 1));
}
//4- Search for a solution
CPSolver s = new CPSolver();
s.read(mod);
s.setVarSetSelector(new MinDomSet(s, s.getVar(vars)));
s.setValSetSelector(new MinEnv());
s.solve();
//5- Print the solution found
for(SetVariable var: vars){
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System.out.println(s.getVar(var).pretty());
}
1.4.3
Example 3: the CycloHexane problem with Choco
Real variables are illustrated on the problem of finding the 3D configuration of a cyclohexane molecule.
//1- Create the problem
CPModel pb = new CPModel();
//2- Create the variable
RealVariable x = Choco.makeRealVar("x", -1.0e8, 1.0e8);
RealVariable y = Choco.makeRealVar("y", -1.0e8, 1.0e8);
RealVariable z = Choco.makeRealVar("z", -1.0e8, 1.0e8);
//3- Create and post the constraints
RealExpressionVariable exp1 = Choco.plus(Choco.mult(Choco.power(y, 2), Choco.plus(1,
Choco.power(z, 2))),
Choco.mult(z, Choco.minus(z, Choco.mult(24, y))));
RealExpressionVariable exp2 = Choco.plus(Choco.mult(Choco.power(z, 2), Choco.plus(1,
Choco.power(x, 2))),
Choco.mult(x, Choco.minus(x, Choco.mult(24, z))));
RealExpressionVariable exp3 = Choco.plus(Choco.mult(Choco.power(x, 2), Choco.plus(1,
Choco.power(y, 2))),
Choco.mult(y, Choco.minus(y, Choco.mult(24, x))));
Constraint eq1 = Choco.eq(exp1, -13);
Constraint eq2 = Choco.eq(exp2, -13);
Constraint eq3 = Choco.eq(exp3, -13);
pb.addConstraint(eq1);
pb.addConstraint(eq2);
pb.addConstraint(eq3);
//4- Search for all solution
CPSolver s = new CPSolver();
s.getConfiguration().putDouble(Configuration.REAL_PRECISION, 1e-8);
s.read(pb);
s.setVarRealSelector(new CyclicRealVarSelector(s));
s.setValRealIterator(new RealIncreasingDomain());
s.solve();
//5- print the solution found
System.out.println("x " + s.getVar(x).getValue());
System.out.println("y " + s.getVar(y).getValue());
System.out.println("z " + s.getVar(z).getValue());
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Chapter 2
The model
The Model, along with the Solver, is one of the two key elements of any Choco program. The Choco
Model allows to describe a problem in an easy and declarative way: it simply records the variables and
the constraints defining the problem.
This section describes the large API provided by Choco to create different types of variables and
constraints.
Note that a static import is required to use the Choco API:
import static choco.Choco.*;
First of all, a Model object is created as follows:
Model model = new CPModel();
In that specific case, a Constraint Programming (CP) Model object has been created.
2.1
Variables
A Variable is defined by a type (integer, real, or set variable), a name, and the values of its domain. When
creating a simple variable, some options can be set to specify its domain representation (e.g., enumerated
or bounded) within the Solver.
The choice of the domain should be considered. The efficiency of the
solver often depends on judicious choice of the domain type.
Variables can be combined as expression variables using operators.
One or more variables can be added to the model using the following methods of the Model class:
model.addVariable(var1);
model.addVariables(var2, var3);
Explictly addition of variables is not mandatory. See Constraint for
more details.
Specific role of variables var can be defined with options: non-decision variables or objective variable;
model.addVariable(Options.V_OBJECTIVE, var4);
model.addVariables(Options.V_NO_DECISION, var5, var6);
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2.1.1
Simple Variables
See Section Variables for details:
IntegerVariable, SetVariable, RealVariable
2.1.2
Constants
A constant is a variable with a fixed domain. An IntegerVariable declared with a unique value is
automatically set as constant. A constant declared twice or more is only stored once in a model.
IntegerConstantVariable c10 = Choco.constant(10);
RealConstantVariable c0dot0 = Choco.constant(0.0);
SetConstantVariable c0_12 = Choco.constant(new int[]{0, 12});
SetConstantVariable cEmpty = Choco.emptySet();
2.1.3
Expression variables and operators
Expression variables represent the result of combinations between variables of the same type made by
operators. Two types of expression variables exist :
IntegerExpressionVariable and RealExpressionVariable.
One can define an expression variable to define an operation, for example:
IntegerVariable v1 = Choco.makeIntVar("v1", 1, 3);
IntegerVariable v2 = Choco.makeIntVar("v2", 1, 3);
IntegerExpressionVariable v1Andv2 = Choco.plus(v1, v2);
To construct expressions of variables, simple operators can be used. Each returns a ExpressionVariable
object:
• Integer : abs, div, ifThenElse, max, min, minus, mod, mult, neg, plus,
power, scalar, sum,
• Real :
cos, minus, mult, plus, power, sin
Note that these operators are not considered as constraints: they do not return a Constraint objet but
a Variable object.
2.1.4
MultipleVariables
These are syntaxic sugar. To make their declaration easier, tree, geost, and scheduling constraints
allow or require to use multiple variables, like TreeParametersObject, GeostObject or TaskVariable.
See also the code examples for these constraints.
2.1.5
Decision/non-decision variables
By default, each variable added to a model is a decision variable, i.e. is included in the default search
strategy. A variable can be stated as a non decision one if its value can be computed by side-effect. To
specify non decision variables, one can
• exclude them from the search strategies (see search strategy for more details);
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• specify non-decision variables (adding Options.V NO DECISION to their options) and keep the default search strategy.
IntegerVariable vNoDec = Choco.makeIntVar("vNoDec", 1, 3, Options.V_NO_DECISION);
Each of these options can also be set within a single instruction for a group of variables, as follows:
IntegerVariable vNoDec1 = Choco.makeIntVar("vNoDec1", 1, 3);
IntegerVariable vNoDec2 = Choco.makeIntVar("vNoDec2", 1, 3);
model.addOptions(Options.V_NO_DECISION, vNoDec1, vNoDec2);
The declaration of a
Options.V NO DECISION.
search
strategy
will
erase
setting
more precise: user-defined/pre-defined, variable and/or value heuristics ?
2.1.6
Objective variable
You can define an objective variable directly within the model, by using option Options.V OBJECTIVE:
IntegerVariable x = Choco.makeIntVar("x", 1, 1000, Options.V_OBJECTIVE);
IntegerVariable y = Choco.makeIntVar("y", 20, 50);
model.addConstraint(Choco.eq(x, Choco.mult(y, 20)));
solver.read(model);
solver.minimize(true);
Only one variable can be defined as an objective. If more than one objective variable is declared,
then only the last one will be taken into account.
Note that optimization problems can be declared without defining an objective variable within the
model (see the optimization example.)
2.2
Constraints
Choco provides a large number of simple and global constraints and allows the user to easily define its
own new constraint. A constraint deals with one or more variables of the model and specify conditions
to be held on these variables. A constraint is stated into the model by using the following methods
available from the Model API:
model.addConstraint(c1);
model.addConstraints(c2,c3);
Adding a constraint automatically adds its variables to the model (explicit declaration of variables addition is not mandatory). Thus, a variable
not involved in any constraints will not be declared in the Solver during
the reading step.
Example:
adding a difference (disequality) constraint between two variables of the model
model.addConstraint(Choco.neq(var1, var2));
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Available options depend on the kind of constraint c to add: they allow, for example, to choose the
filtering algorithm to run during propagation. See Section options and settings for more details, specific
APIs exist for declaring options constraints.
This section presents the constraints available in the Choco API sorted by type or by domain. Related
sections:
• a detailed description (with options, examples, references) of each constraint is given in Section
constraints
• Section user-defined constraint explains how to create its own constraint.
2.2.1
Binary constraints
Constraints involving two integer variables
• eq, geq, gt, leq, lt, neq
• abs, oppositeSign, sameSign
2.2.2
Ternary constraints
Constraints involving three integer variables
• distanceEQ, distanceNEQ, distanceGT, distanceLT
• intDiv, mod, times
2.2.3
Constraints involving real variables
Constraints involving two real variables
• eq, geq, leq
2.2.4
Constraints involving set variables
Set constraints are illustrated on the ternary Steiner problem.
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2.2. CONSTRAINTS
• member, notMember
• eqCard, geqCard, leqCard, neqCard
• eq
• isIncluded, isNotIncluded
• setInter
• setDisjoint, setUnion
• max, min
• inverseSet
• among
• pack
2.2.5
Channeling constraints
The use of a redundant model, based on an alternative set of decision variables, is a frequent technique
to strengthen propagation or to get more freedom to design dedicated search heuristics. The following
constraints allow to ensure the integrity of two redundant models by linking (channeling) variable-value
assignments in the first model to variable-value assignments in the second model:
• boolChanneling bj = 1 ⇐⇒ x = j,
• domainChanneling bj = 1 ⇐⇒ x = j, ∀j,
• inverseChanneling yj = i ⇐⇒ xi = j, ∀i, j,
• inverseChannelingWithinRange yj = i ∧ j ≤ |x| ⇐⇒ xi = j ∧ i ≤ |y|, ∀i, j,
• inverseSet i ∈ sj ⇐⇒ xi = j, ∀i, j,
In the n-queen problem, for example, a domain variable by column indicates the row j to place a queen
in column i. To enhance the propagation, a redundant model can be stated by defining a domain variable
by row indicating the column i. As columns and rows can be interchanged, the same set of constraints
applies to both models, then constraint inverseChanneling is set to propagate between the two models.
int n = 8;
Model m = new CPModel();
IntegerVariable[] queenInCol = new IntegerVariable[n];
IntegerVariable[] queenInRow = new IntegerVariable[n];
for (int i = 0; i < n; i++) {
queenInCol[i] = makeIntVar("QC" + i, 1, n);
queenInRow[i] = makeIntVar("QR" + i, 1, n);
}
for (int i = 0; i < n; i++) {
for (int j = i + 1; j < n; j++) {
int k = j - i;
m.addConstraint(neq(queenInCol[i], queenInCol[j])); // row
m.addConstraint(neq(queenInCol[i], plus(queenInCol[j], k))); // diagonal 1
m.addConstraint(neq(queenInCol[i], minus(queenInCol[j], k))); // diagonal 2
m.addConstraint(neq(queenInRow[i], queenInRow[j])); // column
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m.addConstraint(neq(queenInRow[i], plus(queenInRow[j], k))); // diagonal 2
m.addConstraint(neq(queenInRow[i], minus(queenInRow[j], k))); // diagonal 1
}
}
m.addConstraint(inverseChanneling(queenInCol, queenInRow));
Solver s = new CPSolver();
s.read(m);
s.solveAll();
Channeling constraints are also useful to compose a model made up of two parts as, for example,
in a task-resources assignment problem where some constraints are set on the task set and some other
constraints are set on the resource set.
Reification offers an other type of channeling, between a constraint and a boolean variable representing
its truth value. More complex channeling can be done using reification and boolean operators although
they are less effective. The reified constraint below states b = 1 ⇐⇒ x = y:
IntegerVariable b = Choco.makeBooleanVar("b");
IntegerVariable x = Choco.makeIntVar("x", 0, 10);
IntegerVariable y = Choco.makeIntVar("y", 0, 10);
model.addConstraint(Choco.reifiedConstraint(b, Choco.eq(x, y)));
2.2.6
Constraints in extension and relations
Choco supports the statement of constraints defining arbitrary relations over two or more variables. Such
a relation may be defined by three means:
• feasible table: the list of allowed tuples of values (that belong to the relation),
• infeasible table: the list of forbidden tuples of values (that do not belong to the relation),
• predicate: a method to be called in order to check whether a tuple of values belongs or not to
the relation.
On the one hand, constraints based on tables may be rather memory consuming in case of large domains, although one relation table may be shared by several constraints. On the other hand, predicate
constraints require little memory as they do not cache truth values, but imply some run-time overhead
for calling the feasibility test. Table constraints are thus well suited for constraints over small domains;
while predicate constraints are well suited for situations with large domains.
Different levels of consistency can be enforced on constraints in extension (when selecting an API)
and, for Arc Consistency, different filtering algorithms can be used (when selecting an option). The
Choco API for creating constraints in extension are as follows:
• arc consistency (AC) for binary relations:
feasPairAC, infeasPairAC, relationPairAC
• arc consistency (AC) for n-ary relations:
feasTupleAC, infeasTupleAC, relationTupleAC
• weaker forward checking (FC) for n-ary relations:
feasTupleFC, infeasTupleFC, relationTupleFC
Relations.
A same relation might be shared among several constraints, in this case it is highly recommended to
create it first and then use the relationPairAC, relationTupleAC, or relationTupleFC API on the same
relation for each constraint.
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For binary relations, the following Choco API is provided:
makeBinRelation(int[] min, int[] max, List<int[]>pairs, boolean feas)
It returns a BinRelation giving a list of compatible (feas=true) or incompatible (feas=false) pairs
of values. This relation can be applied to any pair of variables (x1 , x2 ) whose domains are included in
the min/max intervals, i.e. such that:
min[i] ≤ xi .getInf() ≤ xi .getSup() ≤ max[i],
∀i.
Bounds min/max are mandatory in order to allow to compute the opposite of the relation if needed.
For n-ary relations, the corresponding Choco API is:
makeLargeRelation(int[] min, int[] max, List<int[]> tuples, boolean feas);
It returns a LargeRelation. If feas=true, the returned relation matches also the IterLargeRelation
interface which provides constant time iteration abilities over tuples (for compatibility with the GAC
algorithm used over feasible tuples).
LargeRelation r = Choco.makeLargeRelation(min, max, tuples, true);
model.addConstraint(Choco.relationTupleAC(vars, r));
Lastly, some specific relations can be defined without storing the tuples, as in the following example
(TuplesTest extends LargeRelation):
public static class NotAllEqual extends TuplesTest {
public boolean checkTuple(int[] tuple) {
for (int i = 1; i < tuple.length; i++) {
if (tuple[i - 1] != tuple[i]) return true;
}
return false;
}
Then, the NotAllEqual relation is stated as a constraint of a model:
CPModel model = new CPModel();
IntegerVariable[] vars = Choco.makeIntVarArray("v", 3, 1, 3);
model.addConstraint(Choco.relationTupleFC(vars, new NotAllEqual()));
2.2.7
Reified constraints
The truth value of a constraint is a boolean that is true if and only if the constraint holds. To reify a
constraint is to get its truth value.
This mechanism can be used for example to model a MaxCSP problem where the number of satisfied
constraints has to be maximized. It is also intended to give the freedom to model complex constraints
combining several reified constraints, using some logical operators on the truth values, such as in: (x 6=
y) ∨ (z ≤ 9).
Choco provides a generic constraint reifiedConstraint to reify any constraint into a boolean variable
expressing its truth value:
Constraint reifiedConstraint(IntegerVariable b, Constraint c);
Constraint reifiedConstraint(IntegerVariable b, Constraint c1, Constraint c2);
Specific API are also provided to reify boolean constraints:
• reifiedConstraint,
• reifiedAnd, reifiedLeftImp, reifiedNot, reifiedOr, reifiedRightImp,
reifiedXnor, reifiedXor
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Handling complex expressions.
In order to build complex combinations of constraints, Choco also provides a simpler and more direct
API with the following logical meta-constraints taking constraints in arguments:
• and, or, implies, ifOnlyIf, ifThenElse, not, nand, nor
For example, the following expression
((x = 10 ∗ |y|) ∨ (z ≤ 9))
⇐⇒
alldifferent(a, b, c)
could be expressed in Choco by:
Constraint exp = ifOnlyIf( or( eq(x, mult(10, abs(y))), leq(z, 9) ),
alldifferent(new IntegerVariable[]{a,b,c}) );
Such an expression is internally represented as a tree whose nodes are operators and leaves are variables,
constants and constraints. Variables and constants can be combined as ExpressionVariable using
operators (e.g, mult(10,abs(w))), or using simple constraints (e.g., leq(z,9)), or even using global
constraints (e.g, alldifferent(vars)). The language available on expressions currently matches the
language used in the Constraint Solver Competition 2008 of the CPAI workshop.
At the solver level, there exists two different ways to represent expressions:
• by extension: the first way is to handle expressions as constraints in extension. The expression is
then used to check a tuple in a dynamic way just like a n-ary relation that is defined without listing
all the possible tuples. The expression is then propagated using the GAC3rm algorithm. This is
very powerful as arc-consistency is achieved on the corresponding constraints.
• by decomposition: the second way is to decompose the expression automatically by introducing
intermediate variables and possibly the generic reifiedConstraint. By doing so, the level of
pruning decreases but expressions of larger arity involving large domains can be represented.
The way to represent expressions is decided at the modeling level. Representation by extension is the
default. Representation by decomposition can be set instead by:
model.setDefaultExpressionDecomposition(true);
Representation by decomposition can also be controlled individually for some expressions, by setting
option Options.E DECOMP when adding the constraint. For example, the following code tells the solver
to decompose e1 but not e2 :
model.setDefaultExpressionDecomposition(false);
IntegerVariable[] x = makeIntVarArray("x", 3, 1, 3, Options.V_BOUND);
Constraint e1 = or(lt(x[0], x[1]), lt(x[1], x[0]));
model.addConstraint(Options.E_DECOMP, e1);
Constraint e2 = or(lt(x[1], x[2]), lt(x[2], x[1]));
model.addConstraint(e2);
When and how should I use expressions ?
Expressions offer a slightly richer modeling language than the one available via standard constraints.
However, expressions can not be handled as efficiently as constraints that embed a dedicated propagation
algorithm. We therefore recommend you to carefully check that you can not model the expression using
the global constraints of Choco before using expressions.
Expressions represented in extension should be used in the case of complex logical relationships that
involve few different variables, each of small domain, and if arc consistency is desired on those
variables. In such a case, representation in extension can be much more effective than with decomposition.
Imagine the following “crazy” example :
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or( and( eq( abs(sub(div(x,50),div(y,50))),1), eq( abs(sub(mod(x,50),mod(y,50))),2)),
and( eq( abs(sub(div(x,50),div(y,50))),2), eq( abs(sub(mod(x,50),mod(y,50))),1)))
This expression has a small arity: it involves only two variables x and y. Let assume that their domains
has no more than 300 values, then such an expression should typically not be decomposed. Indeed,
arc consistency will create many holes in the domains and filter much more than if the relation was
decomposed.
Conversely, an expression should be decomposed as soon as it involves a large number of variables,
or at least one variable with a large domain.
2.2.8
Global constraints
Choco includes several global constraints. Those constraints accept any number of variables and offer
dedicated filtering algorithms which are able to make deductions where a decomposed model would not.
For instance, constraint alldifferent(a, b, c, d) with a, b ∈ [1, 4] and c, d ∈ [3, 4] allows to deduce that
a and b cannot be instantiated to 3 or 4; such rule cannot be inferred by simple binary constraints.
The up-to-date list of global constraints available in Choco can be found within the Javadoc API. Most
of these global constraints are listed below according to their application fields. Details and examples
can be found in Section Elements of Choco/Constraints.
Value constraints
Constraints that put a restriction on how values can be distributed among a collection of variables. See
also in Global Constraint Catalog: value constraint.
• counting distinct values:
allDifferent, atMostNValue, increasingNValue,
• counting values: among, occurrence, occurrenceMax, occurrenceMin,
globalCardinality,
• indexing values:
nth (element), max, min,
• ordering: sorting, increasingNValue, increasingSum, lex, lexeq, leximin,
lexChain, lexChainEq,
• tuple matching: feasTupleAC, feasTupleFC, infeasTupleAC, infeasTupleFC,
relationTupleAC, relationTupleFC,
• pattern matching:
regular, costRegular, multiCostRegular, stretchPath, tree,
Boolean constraints
Logical operations on boolean expressions. See also in Global Constraint Catalog: boolean constraint.
and, or, clause,
Channeling constraints
Constraints linking two collections of variables (many-to-many) or indexing one among many variables
(one-to-many). See also Section Channeling and in Global Constraint Catalog: channelling constraint.
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• one-to-many:
• many-to-many:
domainChanneling, nth (element), max, min,
inverseChanneling, inverseSet, sorting, pack,
Optimization constraints
Constraints channelling a variable to the sum of the weights of a collection of variable-value assignments.
See also in Global Constraint Catalog: cost-filtering constraint.
• one cost: among, occurrence, occurrenceMax, occurrenceMin, knapsackProblem,
equation, costRegular, tree,
• several costs:
globalCardinality, multiCostRegular,
Packing constraints (capacitated resources)
Constraints involving items to be packed in bins without overlapping. More generaly, any constraints
modelling the concurrent assignment of objects to one or several capacitated resources. See also in Global
Constraint Catalog: resource constraint.
• packing problems:
equation, knapsackProblem, pack (bin-packing),
• geometric placement problems:
• scheduling problems:
• timetabling problems:
geost,
disjoint (tasks) disjunctive, cumulative,
costRegular, multiCostRegular,
Scheduling constraints (time assignment)
Constraints involving tasks to be scheduled over a time horizon. See also in Global Constraint Catalog:
scheduling constraint.
• temporal constraints: disjoint (tasks) precedence, precedenceDisjoint,
precedenceImplied, precedenceReified, forbiddenInterval, tree,
• resource constraints:
2.2.9
cumulative, disjunctive, geost,
Things to know about
Model, Variable
and
Constraint
It is important to know the relation between Models, Variables and Constraints. Variables and Constraints
are build without the help of a Model, so that they can be used natively in different Models. That’s why
one need to add them to a Model, using model.addVariable(Variable var) and model.addConstraint(
Constraint cstr). On a variable addition, this one is added to the list of variables of the model. On
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a constraint addition, the constraint is added to the constraint network AND the constraint and its
associated variables are linked. It means the constraint is now known from its variables, which was not
the case before the addition.
Now, let see a short example: one declares two constraints involving the two same variables, and add
them to five different models. This implies the following references:
• each model points to two constraint and two variables;
• each constraint points the two variables;
• each of the variables points to ten constraints: the very same constraint is considered as different
from a model to another.
This must be kept in mind while you write your program: whether a model is used or not, a shared
variable stores references to the constraints it is involved in.
Do not count on the garbage collector to manage this (even if the finalize() method was declared
for a Model): while a variable references a constraint declared in an obsolete model, neither the
model nor the constraint can be safely destroyed.
A good habit to have is to delete constraints of a model when this one is not used anymore, by
calling model.removeConstraints(). This breaks links between constraint and variable of a model,
prevents large memory consumption and can stabilize performances by reusing Models.
Consider the following code, model2 is faster and consumes less memory than model1 for the same
result.
public static void main(String[] args) {
for (int i = 0; i < 10; i++) {
long t = -System.currentTimeMillis();
model1(99999);
t += System.currentTimeMillis();
System.out.printf("%d ", t);
t = -System.currentTimeMillis();
model2(99999);
t += System.currentTimeMillis();
System.out.printf("%d\n", t);
}
}
private static void model1(int i) {
IntegerVariable v1 = makeIntVar("v1", 0, 10);
IntegerVariable v2 = makeIntVar("v2", 0, 10);
for (int j = 0; j < i; j++) {
CPModel m1 = new CPModel();
m1.addConstraint(eq(v1, v2));
}
}
private static void model2(int i) {
IntegerVariable v1 = makeIntVar("v1", 0, 10);
IntegerVariable v2 = makeIntVar("v2", 0, 10);
CPModel m1 = new CPModel();
for (int j = 0; j < i; j++) {
m1.addConstraint(eq(v1, v2));
m1.removeConstraints();
}
}
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Chapter 3
The solver
The Solver, along with the Model, is one of the two key elements of any Choco program. The Choco
Solver is mainly focused on resolution part: reading the Model, defining the search strategies and the
resolution policy.
To create a Solver, one just needs to create a new object as follow:
Solver solver = new CPSolver();
This instruction creates a Constraint Programming (CP) Solver object.
The solver gives an API to read a model. The reading of a model is compulsory and must be done
after the entire definition of the model.
solver.read(model);
The reading is divided in two parts: variables reading and constraints reading.
3.1
Variables reading
The variables are declared in a model with a given type IntegerVariable, SetVariable, RealVariable
and, possibly, with a given domain type (e.g. bounded or enumerated domains for integer and set variables). When reading the model, the solver iterates over the model variables, then creates the corresponding solver variables and domains data structures according to these types.
Bound variables are related to large domains which are only represented by their lower and upper bounds. The domain is encoded in a
space efficient way and propagation events only concern bound updates.
Value removals between the bounds are therefore ignored (holes are not
considered). The level of consistency achieved by most constraints on
these variables is called bound-consistency.
On the contrary, the domain of an enumerated variable is explicitly
represented and every value is considered while pruning. Basic constraints
are therefore often able to achieve arc-consistency on enumerated variables (except for NP-hard global constraint such as the cumulative constraint). Remember that switching from enumerated variables to bounded
variables decreases the level of propagation achieved by the system.
Model variables and solver variables are distinct objects. Model variables implement the
Variable interface and solver variables implement the Var interface. A model variable is defined by
an abstract representation of its initial domain, while a solver variable encapsulates a concrete representation of the domain, and maintains its current state throughout the search. Hence, one cannot access
a variable value directly from a model variable but one can from its corresponding solver variable. The
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solver variables are anonymous but can be accessed from the corresponding model variables using the
Solver API getVar(Variable v) and getVar(Variable... v).
3.1.1
from IntegerVariable toIntDomainVar
For integer variables, the solver IntDomainVar is counterpart to the model IntegerVariable. Methods getVar(IntegerVariable var) and getVar(IntegerVariable... vars) of Solver return the
objects IntDomainVar and IntDomainVar[] respectively corresponding to var and vars:
IntegerVariable x = Choco.makeEnumIntVar("x", 1, 100); // model variable
IntDomainVar xOnSolver = solver.getVar(x); // solver variable
The state of an IntDomainVar can be accessed using these main public methods:
IntDomainVar API
description
hasEnumeratedDomain()
checks if the domain type is enumerated or bounded
getInf()
returns the current lower bound of the variable
getSup()
returns the current upper bound of the variable
getVal()
returns the value of the variable if it is currently instantiated
isInstantiated()
checks if the domain is currently reduced to a singleton
canBeInstantiatedTo(int v)
checks if value v currently belongs to the domain of the variable
getDomainSize()
returns the current size of the domain
The data structure representing the current domain within the IntDomainVar object depends on the
domain type (bounded, enumerated, boolean, constant, etc.) of the model variable. See advanced uses
for more informations on IntDomainVar.
3.1.2
from SetVariable to SetVar
For set variables, the solver SetVar is counterpart to the model SetVariable. Methods getVar(SetVariable
var) and getVar(SetVariable... vars) of Solver return the objects SetVar and SetVar[] respectively corresponding to var and vars:
SetVariable x = Choco.makeBoundSetVar("x", 1, 40); // model variable
SetVar xOnSolver = solver.getVar(x); // solver variable
Note that a set variable on integer values between 1 and n may have 2n possible values, corresponding
to every possible subsets of {1, 2, . . . , n}. Hence, the domain of a SetVar is encoded by these bounds
only: the lower bound, called the kernel, is the intersection of all possible set values, and the upper
bound, called the envelope, is the union of all possible set values. Furthermore, a SetVar encapsulates
an IntDomainVar representing the cardinality of the set variable. The domain type of this variable
(enumerated or bounded) depends on the option given at the construction of the SetVariable.
The state of a SetVar can be accessed through these main public methods: Warning: Envelope is
(french) spelled with two ’p’ in the method name.
SetVar API
description
getCard()
returns the current cardinality (an IntDomainVar object)
isInDomainKernel(int v)
checks if value v belongs to the current kernel
isInDomainEnveloppe(int v)
checks if value v belongs to the current envelope
getDomain()
returns the current domain (a SetDomain object). Iterators on envelope
or kernel can then be called
getKernelDomainSize()
returns the current size of the kernel
getEnveloppeDomainSize()
returns the current size of the envelope
getEnveloppeInf()
returns the current smallest value of the envelope
getEnveloppeSup()
returns the current largest value of the envelope
getKernelInf()
returns the current smallest value of the kernel
getKernelSup()
returns the current largest value of the kernel
getValue()
returns the set value as a table of integers int[] when the variable is
currently instantiated (kernel=envelope)
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See advanced uses for more informations on SetVar.
3.1.3
from RealVariable to RealVar
Real variables are still under development but can be used to solve toy
problems such as small systems of equations.
For real variables, the solver RealVar is counterpart to the model RealVariable. Methods getVar(RealVariable
var) and getVar(RealVariable... vars) of Solver return the objects RealVar and RealVar[] respectively corresponding to var and vars:
RealVariable x = Choco.makeRealVar("x", 1.0, 3.0); // model variable
RealVar xOnSolver = solver.getVar(x); // solver variable
Continuous variables are useful for non linear equation systems which are encountered in physics for
example. The state of a RealVar can be accessed through these main public methods:
RealVar API
description
getInf()
returns the current lower bound of the variable (double)
getSup()
returns the current upper bound of the variable (double)
isInstantiated()
checks if the domain is reduced to a canonical interval. A canonical
interval indicates that the domain has reached the precision given by the
user or the solver
See advanced uses for more informations on RealVar.
3.1.4
from TaskVariable to TaskVar
For task variables, the solver TaskVar is counterpart to the model TaskVariable. Methods getVar(TaskVariable
var) and getVar(TaskVariable... vars) of Solver return the objects TaskVar and TaskVar[] respectively corresponding to var and vars:
TaskVariable x = Choco.makeTaskVar("x", 0, 123, 18); // model variable
TaskVar xOnSolver = solver.getVar(x); // solver variable
Task variables help at formulating scheduling problems where one has to determine the starting and
ending time of a task. A task variable aggregates three integer variable: start, end, and duration and
the implicit constraint start+duration=end.
To complete. The state of a TaskVar can be accessed through these main public methods:
TaskVar API
description
isInstantiated()
checks if the three time integer variables are instantiated
See advanced uses for more informations on TaskVar.
3.2
Constraints reading
Once the solver variables are created when reading the model, the solver then iterates over the constraints of the model, and creates the solver SConstraint objects by calling method makeConstraint
of the ConstraintManager object associated to the model constraint type. At this step, auxiliary solver
variables and constraints may be generated. The created constraints are then added to the internal
constraint network.
Each solver constraint encapsulates a filtering algorithm which is called, during the search, when a
propagation step occurs or when an external event (e.g., value removal or bound modification) happens
on some variable of the constraint.
One can access the Solver representation of a Model constraint, using the Solver API getCstr(
Constraint c).
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3.3
Solve a problem
Table below presents the different API offered by Solver to launch the problem resolution. All these
methods return a Boolean object standing for the problem feasibility status of the solver:


Boolean.TRUE
Boolean.FALSE


null
if at least one feasible solution has been computed,
if the problem is proved to be infeasible,
otherwise, i.e. when a search limit has been reached before.
Solver API
solve() or solve(false)
description
runs backtracking until reaching a first feasible solution (returns Boolean.TRUE) or the proof of infeasibility (returns
Boolean.FALSE) or a search limit (returns null).
nextSolution()
Can only be called after a solve() or a nextSolution()
call that has returned Boolean.TRUE. Runs backtracking,
from the solution leaf reached by the previous solve() or
nextSolution() call, until reaching a new feasible solution
(returns Boolean.TRUE), or proving no such new solution exists
(returns Boolean.FALSE), or reaching a search limit (returns
null).
isFeasible()
Returns the feasibility status of the solver.
solveAll() or solve(true)
Runs backtracking until computing all feasible solutions, or until proving infeasibility (returns Boolean.FALSE) or until reaching a search limit (returns Boolean.TRUE if at least one first
solution was computed, and null otherwise).
maximize(Var obj, boolean restart),
maximize(boolean restart)
Runs branch-and-bound until reaching a feasible solution that
is proved to maximize objective obj, or until proving infeasibility (returns Boolean.FALSE) or until reaching a search limit (returns Boolean.TRUE if at least one first solution was computed,
and null otherwise). It proceeds by successive improvements
of the best solution found so far: each time a feasible solution
is found at a leaf of the tree search, then the search proceeds
for a new solution with a greater objective, until it proves that
no such improving solution exists. Parameter restart is a
boolean indicating whether the search continues from the solution leaf with a backtrack (if set to false) or if it is restarted
from the root node (if set to true).
minimize(Var obj, boolean restart),
minimize(boolean restart)
similar to maximize but for computing a feasible solution that
is proved to minimize objective obj.
The following API are also useful to manipulate a Solver object:
Solver API
description
propagate()
Launchs propagation by running, in turn, the domain reduction algorithms of the constraints until it reaches a fix point.
Throws a ContradictionException when a contradiction is
detected, i.e. a variable domain is emptied. This method is
called at each node of the tree search constructed by the solving methods above.
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3.4. STORING ET RESTORING SOLUTIONS
3.4
Storing et restoring solutions
By default, the last solution found is stored and accessible once the search has ended.
Store solutions. To store the solutions found during the resolution, one just needs to override the
solution pool capacity (which is set to 1 by default).
int size = Integer.MAX_VALUE;
solver.getConfiguration().putInt(Configuration.SOLUTION_POOL_CAPACITY, size);
solver.solveAll();
ISolutionPool pool = solver.getSearchStrategy().getSolutionPool();
Setting size to Integer.MAX_VALUE means “store every solution”, otherwise you can set a finite value,
for example 10 and the last 10 solutions will be stored.
A Solution object is made of arrays of values: an array of values of IntVar, another one for values
of SetVar and the last one for the values of RealVar. One can retrieve the value of a variable by calling
getXXValue(int idx), where XX is the type of variable and idx is its index within the solver:
solution.getIntValue(solver.getIntVarIndex(ivar));
solution.getSetValue(solver.getSetVarIndex(svar));
solution.getRealValue(solver.getRealVarIndex(fvar));
And to make the link between a variable in a solution, simply retrieve the index of the variable in
the solver:
int q0 = solver.getIntVarIndex(solver.getVar(vars[0])); // vars is an array of
IntegerVariable
solution.getIntValue(q0);
Restore solutions. A solution can be restored within a solver. But to do that, some precautions must
be made: before starting the search, the state of the solver must be backed up.
int rootworld = solver.getEnvironment().getWorldIndex();
solver.worldPush();
solver.solveAll();
And then, to restore, first restore the root world and backup another world:
solver.worldPopUntil(rootworld); // restore the original state, where domains were as
declared (not yet instantiated)
solver.worldPush(); // backup the current state of the solver, to allow other solution
restoration
solver.restoreSolution(solution); // restore the solution
// do something
If you want to restore more than one solution, you just apply this 3 steps.
3.5
Search Strategy
A key ingredient of any constraint approach is a clever search strategy. In backtracking or branchand-bound approaches, the search is organized as an enumeration tree, where each node corresponds to
a subspace of the search, and each child node is a subdivision of its father node’s space. The tree is
progressively constructed by applying a series of branching strategies that determine how to subdivise
space at each node and in which order to explore the created child nodes. Branching strategies play the
role of achieving intermediate goals in logic programming.
This section presents how to define your own search strategy in Choco.
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Standard backtracking or branch-and-bound approaches in constraint
programming develop the enumeration tree in a Depth-First Search
(DFS) manner:
1. evaluate a node: run propagation
2. if a failure occurs or if the search space cannot be separated then
backtrack : evaluate the next pending node
3. otherwise branch: divide the search space and evaluate the first
child node.
With Choco, the search process of the CPSolver does not currently allow
to explore the tree in a different manner, using Best-First Search for
example.
In addition, the common way of dividing the search space in CP-based
backtracking/B&B algorithms is to assign a variable to a value or to
forbid this assignment. Choco provides such a branching strategy and
the tools to easily customize the variable and value selection heuristics
within this strategy. However, Choco makes possible to implement more
complex branching strategies (e.g. constraint branching or dichotomy
branching).
3.5.1
Overriding the default search strategy
Branching, variable selection and value selection strategies. Basically, a search strategy in
Choco is a composition of branching strategy objects, each defined on a given set of decision variables.
The most common branching strategies are based on the assignment of a selected variable to one or
several selected values (one assignment in each branch).
Branching strategies apply to Solver variables (not Model variables).
The variable and value selection heuristics can be defined separately in their own objects: a variable
selector and a value selector or a value iterator. Branching strategy AssignVar, for example, can simply
be customized via these embedded objects: the variable is first selected, then a value in the variable
domain is selected. The two following instructions both create a n-ary branching strategy (AssignVar)
selecting an integer decision variable of minimum domain size (MinDomain variable selector) and assigning
it successively, in each branch, to one of its domain value, selected in increasing order (IncreasingDomain
value iterator or MinVal value selector).
new AssignVar(new MinDomain(solver), new IncreasingDomain());
new AssignVar(new MinDomain(solver), new MinVal());
Note that this usual strategy is pre-defined in BranchingFactory, and may then also be declared as
follows:
BranchingFactory.minDomMinVal(solver);
Sometimes, the choice of the variable may also depend on the choice of the value, or it may require
specific computations before or after branching. In this case, the variable selection heuristic can directly
be implemented in the branching strategy object, like e.g. in DomOverWDegBranchingNew. Both
variable and value selection heuristics can be implemented directly within the branching strategy, like
e.g in ImpactBasedBranching.
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3.5. SEARCH STRATEGY
Default strategies. When no search strategy is specified, default search strategies apply to all the
decision variables of the solver. These strategies vary according to the variable types:
Variable type
Default strategy
Set
AssignSetVar + MinDomainSet + MinEnv
Integer
DomOverWDegBranchingNew +IncreasingDomain
Real
AssignInterval + CyclicRealVarSelector+ RealIncreasingDomain
If the model has decision variables of different types, then these default branchings are evaluated in this
order: first, the set decision variables are considered until they are all instantiated, then branching occurs
on the pool of integer decision variables, and last on the pool of real decision variables.
Decision variables. Branchings apply to decision variables only. A branching can occur (i.e. the tree
node can be separated according to this strategy) if and only if there exists a decision variable in its
scope that is still not instantiated. The non-decision variables are also called implied variables because
it is expected that, all variables – including these – will be instantiated (i.e. they will form a solution) by
propagation as soon as all the decision variables will be instantiated. Consider for example, a problem
with two sets of variables x and y linked by channeling or by some implication x = S =⇒ y = T then
the x variables can be set as the decision variables, the y will be instantiated by side-effect.
By default, every solver variable belongs to the pool of decision variables, unless:
• it corresponds to a model variable created with flag Options.V NO DECISION;
• or it is internally created by the solver (e.g. when reading some model constraint) and explicitely
excluded from the pool;
• or the default branching strategies are overriden and the variable does not belong to the scope of
one of the strategies specified by the user.
The scope of a branching strategy is defined at the creation of the strategy. For example,
new AssignSetVar(new MinDomSet(solver, solver.getVar(svars)), new MinEnv()));
defines a branching strategy that only applies to the solver variables corresponding to the model set
variables svars (even if they were defined with flag Options.V NO DECISION).
Most branching strategies may be declared without specifying their scope. In this case, they apply
to all the solver decision variables of the right type. For example, the branching strategy
new AssignSetVar(new MinDomSet(solver), new MinEnv()));
now applies to all the solver decision set variables.
If the default strategies of the solver are overridden by this strategy alone, then all other integer and
real variables will automatically be removed from the decision pool: one has then to ensure that the
instantiation of the set variables alone defines a complete solution. If it is not the case, the branching
strategy must be combined with additional branching strategies holding on the remaining unimplied
variables.
If the default strategies are overriden, then the pool of decision variables
is overriden by the union of the scopes of the user-specified branching
strategies.
As the branching strategies are evaluated sequentially, a variable may belong to the scope of two different
strategies, but it will only be considered by the first strategy, unless this first (user-defined) strategy let
the variable un-instantiated.
Overriding the default search strategies.
using the following API of Solver:
A branching strategy is added to the solver, as a goal,
void addGoal(AbstractIntBranchingStrategy branching);
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This method must be called on the solver object before calling the solving method. The initial list of goals
is empty. If goals are specified, they are added to the list in the order of their declaration. Otherwise,
the list is initialized with the default goals (in the order: set, integer, real).
When one wants to relaunch the search, the list of goals of the solver can previously be reset using
the following instruction:
solver.clearGoals();
Complete example. The following example adds four branching objects to solver s. The first two
branchings are both AssignVar strategies using different variable/value selection heuristics and applied
to different scopes: the integer variables vars1 and vars2, respectively. The third strategy applies to the
set variables svars. The last random strategy applies to all the integer decision variables of the solver.
s.addGoal(BranchingFactory.minDomMinVal(s, s.getVar(vars1)));
s.addGoal(new AssignVar(new DomOverDeg(s, s.getVar(vars2)), new DecreasingDomain()));
s.addGoal(new AssignSetVar(new MinDomSet(s, s.getVar(svars)), new MinEnv()));
s.addGoal(BranchingFactory.randomIntSearch(s, seed));
s.solve();
The goals are evaluated in this order: first, variables vars1 are considered until they are all instantiated,
then branching occurs on variables vars2, then on variables svars. Finally, a random strategy is applied
to all the integer decision variables of the solver that are not already instantiated, thereby excluding
variables vars1 and vars2.
3.5.2
Pre-defined search strategies
This section presents the strategies available in Choco. These objects are also detailed in Part Elements
of Choco. See Chapter advanced uses for a description of how to write search strategies in Choco.
Branching strategy defines the way to branch from a tree search node.
The branching strategies currently available in Choco are the following:
AssignInterval, AssignOrForbidIntVarVal, AssignOrForbidIntVarValPair,
AssignSetVar, AssignVar, DomOverWDegBranchingNew, DomOverWDegBinBranchingNew,
ImpactBasedBranching, PackDynRemovals, SetTimes, TaskOverWDegBinBranching.
They implement interface BranchingStrategy.
Variable selector defines the way to choose a non instantiated variable on which the next decision
will be made.
The variable selectors currently available in Choco are the following:
• implementing interface VarSelector<IntDomainVar>:
CompositeIntVarSelector, LexIntVarSelector, MaxDomain, MaxRegret,
MaxValueDomain, MinDomain, MinValueDomain, MostConstrained,
RandomIntVarSelector, StaticVarOrder
• implementing interface VarSelector<SetVar>:
MaxDomSet, MaxRegretSet, MaxValueDomSet, MinDomSet, MinValueDomSet,
MostConstrainedSet, RandomSetVarSelector, StaticSetVarOrder
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3.5. SEARCH STRATEGY
• implementing interface VarSelector<RealVar>:
CyclicRealVarSelector
Value iterator
Once the variable has been choosen, the solver has to compute its value. The first way to do it is to
schedule all the values once and to give an iterator to the solver.
The value iterators currently available in Choco are the following:
• implementing interface ValIterator<IntDomainVar>:
DecreasingDomain, IncreasingDomain
• implementing interface ValIterator<RealVar>:
RealIncreasingDomain
Value selector
The second way to do it is to compute the next value at each call.
The integer value selector currently available in Choco are the following:
• implementing interface ValSelector<IntDomainVar>:
– MaxVal, MidVal, MinVal
– BestFit, CostRegularValSelector, FCostRegularValSelector,
RandomIntValSelector,
• implementing interface ValSelector<SetVar>:
MinEnv, RandomSetValSelector
3.5.3
Why is it important to define a search strategy ?
The search strategy should not be under-estimatimated!! A well-suited
search strategy can reduce: the execution time, the number of expanded
nodes, the number of backtracks.
Let see that small example:
Model m = new CPModel();
int n = 1000;
IntegerVariable var = Choco.makeIntVar("var", 0, 2);
IntegerVariable[] bi = Choco.makeBooleanVarArray("b", n);
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m.addConstraint(Choco.eq(var, Choco.sum(bi)));
Solver badStrat = new CPSolver();
badStrat.read(m);
badStrat.addGoal(
new AssignVar(
new MinDomain(badStrat),
new IncreasingDomain()
));
badStrat.solve();
badStrat.printRuntimeStatistics();
Solver goodStrat = new CPSolver();
goodStrat.read(m);
goodStrat.addGoal(
new AssignVar(
new MinDomain(goodStrat, goodStrat.getVar(new IntegerVariable[]{var})),
new IncreasingDomain()
));
goodStrat.solve();
goodStrat.printRuntimeStatistics();
This model ensures that var = b0 + b1 + . . . + b1000 where var is an integer variable with a small
domain [0, 2] and bi are binary variables. No deduction arose from the propagation here, so a fix point is
reached at the beginning of the search. A branching decision has to be taken, by selecting a variable and
the first value to assign to it. Using the first strategy, the solver will find a solution after creating 1001
nodes: it iterates over all the variables, starting by assigning the 1000 binary variables bi (according to
the MinDomain variable selector) to 0 (according to the IncreasingDomain value iterator), variable var
is fixed to 0 at the very last propagation. The second strategy finds the same solution with only two
nodes: after branching first on var = 0, propagation immediately fixes all the binary variables to 0.
3.5.4
Restarts
Restart means stopping the current tree search, then starting a new tree search from the root node.
Restarting makes sense only when coupled with randomized dynamic branching strategies ensuring that
the same enumeration tree is not constructed twice. The branching strategies based on the past experience of the search, such as DomOverWDegBranching, DomOverWDegBinBranching and ImpactBasedBranching,
are more accurate in combination with a restart approach.
Unless the number of allowed restarts is limited, a tree search with restarts is not complete anymore.
It is a good strategy, though, when optimizing an NP-hard problem in a limited time.
Restarts can be set using the following API available on the Solver:
setGeometricRestart(int base, double grow);
setGeometricRestart(int base, double grow, int restartLimit);
It performs a search with restarts controlled by the number of backtracks. Parameter base indicates the
maximal number of backtracks allowed in the first search tree. Once this limit is reached, a restart occurs
and the search continues until base*grow backtracks are done, and so on. After each restart, the limit
number of backtracks is increased by the geometric factor grow. restartLimit states the maximum
number of restarts.
CPSolver s = new CPSolver();
s.read(model);
s.setGeometricRestart(14, 1.5d);
s.setFirstSolution(true);
s.generateSearchStrategy();
s.attachGoal(new DomOverWDegBranching(s, new IncreasingDomain()));
s.launch();
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3.6. LIMITING SEARCH SPACE
The Luby’s restart policy is an alternative to the geometric restart policy, and can be defined using
the following API available on the Solver:
setLubyRestart(int base);
setLubyRestart(int base, int grow);
setLubyRestart(int base, int grow, int restartLimit);
It performs a search with restarts controlled by the number of backtracks. The maximum number of
backtracks allowed at a given restart iteration is given by base multiplied by the Las Vegas coefficient at
this iteration. The sequence of these coefficients is defined recursively on its prefix subsequences: starting
from the first prefix 1, the (k+1)-th prefix is the k-th prefix repeated grow times and immediately followed
by coefficient growk .
• the first coefficients for grow=2 : [1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 4, 8, 1,...]
• the first coefficients for grow=3 : [1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 9,...]
CPSolver s = new CPSolver();
s.read(model);
s.setLubyRestart(50, 2, 100);
s.setFirstSolution(true);
s.generateSearchStrategy();
s.attachGoal(new DomOverWDegBranching(s, new IncreasingDomain()));
s.launch();
3.6
Limiting Search Space
The Solver class provides ways to limit the tree search controlled by different criteria. These limits have
to be specified before the resolution. They are updated and checked each time a new node is created.
Once a limit is reached, the search stops even if no solution is found.
time limit: concerns the elapsed time from the beginning of the search (i.e. from the call to a resolution
method). A time limit is set using the Solver API setTimeLimit(int timeLimit), where timeLimit
is in milliseconds. getTimeCount() returns the total solving time.
node limit: concerns the number of opened nodes. A node limit is set using the Solver API setNodeLimit
(int nodeLimit). getNodeCount() returns the total number of explored nodes.
backtrack limit: concerns the number of performed backtracks. A backtrack limit is set using the
Solver API setBackTrackLimit(int backtrackLimit). getBackTrackCount() return the total number of backtracks.
fail limit: concerns the number of contradiction encountered. A fail limit is set using the Solver API
setFailLimit(int failLimit). getFailCount() returns the total number of failures. By default, the
failure count is recorded, one should call monitorFailLimit(true) to activate it.
Define all these notions more precisely and add an example.
3.7
Logging the search
A logging class is available to produce trace statements throughout search: ChocoLogging.
3.7.1
Architecture
Choco logging system is based on the java.util.logging package and located in the package common.
logging. Most Choco abstract classes or interfaces propose a static field LOGGER. The following figures
present the architecture of the logging system with the default verbosity.
The shape of the node depicts the type of logger:
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Figure 3.1: Logger Tree with the default verbosity
• The house loggers represent private loggers. Do not use these loggers directly because their level
are low and all messages would always be displayed.
• The octagon loggers represent critical loggers. These loggers are provided in the variables, constraints and search classes and could have a huge impact on the global performance.
• The box loggers are provided for dev and users.
The color of the node gives its logging level with DEFAULT verbosity: Level.FINEST (gold), Level.INFO
(orange), Level.WARNING (red).
3.7.2
Verbosities and messages
The verbosity level of the solver can be set by the following static method :
// Before the resolution
ChocoLogging.toVerbose();
//... resolution declaration
solver.solve();
// And after the resolution
ChocoLogging.flushLogs();
The following table summarizes the verbosities available in choco:
• OFF – level 0: Disable logging.
Usage : ChocoLogging.setVerbosity(Verbosity.OFF)
• SILENT – level 1: Display only severe messages.
Usage :
– ChocoLogging.toSilent()
– ChocoLogging.setVerbosity(Verbosity.SILENT)
• DEFAULT – level 2: Display informations on final search state.
– ON START
** CHOCO : Constraint Programming Solver
** CHOCO v2 .1.1 ( April , 2009) , Copyleft ( c ) 1999 -2010
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3.7. LOGGING THE SEARCH
– ON COMPLETE SEARCH:
- Search completed [ Maximize
: {0} ,]
[ Minimize
: {1} ,]
Solutions
: {2} ,
Times ( ms )
: {3} ,
Nodes
: {4} ,
Backtracks
: {5} ,
Restarts
: {6}.
brackets [line] indicate line is optional,
Maximize –resp. Minimize– indicates the best known value before exiting of the objective
value in maximize() –resp. minimize()– strategy.
– ON COMPLETE SEARCH WITHOUT SOLUTIONS :
- Search completed - No solutions
[ Maximize
: {0} ,]
[ Minimize
: {1} ,]
Solutions
: {2} ,
Times ( ms )
: {3} ,
Nodes
: {4} ,
Backtracks
: {5} ,
Restarts
: {6}.
brackets [line] indicate line is optional,
Maximize –resp. Minimize– indicates the best known value before exiting of the objective
value in maximize() –resp. minimize()– strategy.
– ON INCOMPLETE SEARCH:
- Search incompleted - Exiting on limit reached
Limit
: {0} ,
[ Maximize
: {1} ,]
[ Minimize
: {2} ,]
Solutions
: {3} ,
Times ( ms )
: {4} ,
Nodes
: {5} ,
Backtracks
: {6} ,
Restarts
: {7}.
brackets [line] indicate line is optional,
Maximize –resp. Minimize– indicates the best known value before exiting of the objective
value in maximize() –resp. minimize()– strategy.
Usage :
– ChocoLogging.toDefault()
– ChocoLogging.setVerbosity(Verbosity.DEFAULT)
• VERBOSE – level 3: Display informations on search state.
– EVERY X (default=1000) NODES:
- Partial search - [Objective : {0}, ]{1} solutions, {2} Time (ms), {3} Nodes, {4}
Backtracks, {5} Restarts.
Objective indicates the best known value.
– ON RESTART :
- Restarting search - {0} Restarts.
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Usage :
– ChocoLogging.toVerbose()
– ChocoLogging.setVerbosity(Verbosity.VERBOSE)
• SOLUTION – level 4: display all solutions.
– AT EACH SOLUTION:
- Solution #{0} found. [Objective: {0}, ]{1} Solutions, {2} Time (ms), {3} Nodes, {4}
Backtracks, {5} Restarts.
X_1:v1, x_2:v2...
Usage :
– ChocoLogging.toSolution()
– ChocoLogging.setVerbosity(Verbosity.SOLUTION)
• SEARCH – level 5: Display the search tree.
– AT EACH NODE, ON DOWN BRANCH:
...[w] down branch X==v branch b
where w is the current world index, X the branching variable, v the branching value and b the
branch index. This message can be adapted on variable type and search strategy.
– AT EACH NODE, ON UP BRANCH:
...[w] up branch X==v branch b
where w is the current world index, X the branching variable, v the branching value and b the
branch index.
Usage :
– ChocoLogging.toSearch()
– ChocoLogging.setVerbosity(Verbosity.SEARCH)
• FINEST – level 6: display all logs.
Usage : ChocoLogging.setVerbosity(Verbosity.FINEST)
More precisely, if the verbosity level is greater than DEFAULT, then the verbosity levels of the model
and of the solver are increased to INFO, and the verbosity levels of the search and of the branching are
slightly modified to display the solution(s) and search messages.
3.7.3
Basic settings
Note that in the case of a verbosity greater or equals to toVerbose(), the regular search information
step is set to 1000, by default. You can change this value, using:
ChocoLogging.setEveryXNodes(20000);
Note that in the case of verbosity toSearch(), trace statements are printed up to a maximal depth
in the search tree. The default value is set to 25, but you can change the value of this threshold, say to
10, with the following setter method:
ChocoLogging.setLoggingMaxDepth(10);
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3.8. CLEAN A SOLVER
3.8
Clean a Solver
Although it is simple and secure to create new instance of Solver, sometimes it is more obvious and
efficient to reuse a Solver. It is recommended to reuse instance of Solver when a problem is being solved
more than once without deep modifications between two resolutions:
• a problem is being resolved with different search strategies,
• a problem is being modified (by adding or removing constraints) through multiple resolutions.
Reusing a Solver must be prepared and well thought out. A backup of
the initial state of the Solver may be necessary to recover initial domains
and constraints internal structures. In the case where new constraints
are created and added, this must be done manipulating Solver objects
(SConstraint for example). So, this required knowledge in advanced uses
of Solver.
What are the methods to clean up a Solver ?
• reset the search strategy : this is done by calling resetSearchStrategy() on a CPSolver. A call to
this method clears the defined branching strategies (safely removes previous ones) and sets the
current search strategy to null. User’s defined branching strategies must be defined again. User’s
defined limits will be reset for the next search.
• cancel restarts : this is done by calling cancelRestarts(Solver solver) on RestartFactory. Sets the
restarts parameters to the default ones. This must be done if a restart strategy has been declared.
3.8.1
What about simply calling
solver.clear()?
A call to solver.clear() will reset every internal structures of a Solver: it clears the variable list, the
constraint list, the environment, the propagation engine, the model read, etc. This set the solver in the
same state as it was just after its creation.
3.8.2
Things to know about
Solver
reusability
There are few things to know:
• a variable cannot be removed from a Solver,
• a statically posted constraint (added using solver.postCut(SConstraint c)) can be removed at any
time using void eraseConstraint(SConstraint c),
• a dynamically posted constraint (added using solver.post(SConstraint c)) can be removed at root
node using void eraseConstraint(SConstraint c).
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Chapter 4
Advanced uses of Choco
4.1
Environment
Environment is a central object of the backtracking system. It defines the notion of world. A world
contains values of storable objects or operations that permit to backtrack to its state. The environment
pushes and pops worlds.
There are primitive data types (IStateBitSet, IStateBool, IStateDouble, IStateInt, IStateLong)
and objects data types (IStateBinarytree, IStateIntInterval, IStateIntProcedure, IStateIntVector,
IStateObject, IStateVector).
There are two different environments: EnvironmentTrailing and EnvironmentCopying.
4.1.1
Copying
In that environment, each data type is defined by a value (primitive or object) and a timestamp. Every
time a world is pushed, each value is copied in an array (one array per data type), with finite indice.
When a world is popped, every value is restored.
4.1.2
Trailing
In that environment, data types are defined by its value. Every operation applied to a data type is
pushed in a trailer. When a world is pushed, the indice of the last operation is stored. When a world
is popped, these operations are popped and unapplied until reaching the last operation of the previous
world.
Default one in CPSolver
4.1.3
Backtrackable structures
to complete
4.2
How does the propagation engine work ?
Once the Model and Solver have been defined, the resolution can start. It is based on decisions and
filtering orders, this is the propagation engine. In this part, we’re going to present how the resolution is
guided in Choco.
A resolution instruction (solve(), solveAll(), maximize(...) or minimize(...)) always starts by setting options based on resolution policy, then generates the search strategies and ends by running the
search loop.
4.2.1
How does a search loop work ?
The search loop is the conductor of the engine. It goes down and up in the branches in order to cover
the tree search, call the filtering algorithm, etc. Here is the organigram of the search loop.
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IN
initial
propagation
true
unsatisfied?
unknown
restart
has next true
branching
object?
false
{luby, walsh, none}
?
select branching object
false
true
is a
solution?
- no goods recording
- restore ROOT node
- post dynamic cut
false
false
- backup state
- apply choice
ERROR
- restore previous state
- post dynamic cut
- apply opposite choice
propagate
true
record solution
propagate
unknown
propagate
continue
search?
unknown
true
unsatisfied?
true
has reached
root node?
unsatisfied?
true
false
true
unknown
unsatisfied?
true
true
restart?
false
OUT
Figure 4.1: Organigram of the search loop
Basically, the search loop is divided in 5 steps: INITIAL PROPAGATION (highlighted in red), OPEN NODE
(highlighted in green), DOWN BRANCH (highlighted in violet), UP BRANCH (highlighted in orange) and RESTART
(highlighted in blue).
4.2.2
Propagate
The main and unique PropagationEngine of Choco is ChocoEngine. This engine stores events occurring on
variables, variable events, and specific calls to constraint filtering algorithm, constraint events, in order
to reach a fix point or to detect contradictions. Events are stored in queues (FIFO).
On a call to Solver.propagate() or during a resolution step, the consistency of a model is computed:
stored events are popped and propagated (apply side-effects). The propagation of a single event can
create new ones, feeding the system until fix point or contradiction.
IN
at least
one variable
event ?
at least
one constraint
event ?
false
true
false
OUT
true
pop event + propagate
false
contradiction?
true
error
Figure 4.2: Organigram of the propagation loop
If the propagation of an event leads to a contradiction, the propagation engine stop the process. In
both case, the search loop take up with the new state.
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4.2. HOW DOES THE PROPAGATION ENGINE WORK ?
Seven priorities
Before going further, it is important to know that events declare a parameter named priority. The priority
of a constraint’s event depends on the constraint priority (required in the super class constructor). And
the priority of a variable’s event is given by the maximum priority of the variable’s constraints.
To each priority corresponds a rank. In the propagation engine, there are as many queues as ranks.
During the propagation loop, the rank is used to push the event on the corresponding queue. Events are
always popped from the smallest ranked queue to the largest ranked queue.
There are seven priorities : UNARY, BINARY, TERNARY, LINEAR, QUADRATIC, CUBIC and VERY_SLOW, each of
these qualifies the ”cost” of a constraint (related to its internal filtering algorithm).
By default, priorities and ranks are defined as follow:
Priority
unary
binary
ternary
linear
quadratic
cubic
very slow
Rank
1
2
3
4
5
6
7
Although the priority of a constraint cannot be changed, the rank of priority can be reconsider by
setting another value to Configuration.VEQ_ORDER or Configuration.CEQ_ORDER.
Configuration conf = new Configuration();
conf.putInt(Configuration.VEQ_ORDER, 7654321); // default value is 1234567
conf.putInt(Configuration.CEQ_ORDER, 1111744); // default value is 1234567
Solver solver = new CPSolver(conf);
In that example, priorities ranks are reversed for the variable queues, and totally rearranged for the
constraint queue.
Constraint event
At the very beginning of resolution, when constraint filtering algorithms have not been called once, a
call to the awake() method is planned by posting constraint event to the propagation engine. For some
expensive constraints (like Geost), a call to the main filtering algorithm (described in propagate()) can
be added during the resolution by posting a constraint event to the propagation engine, in that case a
call to the propagate() method is planned. This event is added to the list of constraint events to be
treated by the propagation engine.
Variable event
The resolution goal is to instantiate variables in order to find solutions. Instantiation of a variable is
done applying modification on its domain. Each time a modification is applied on a domain, an event is
posted, storing informations about the action done (event type, variable, values, etc.). This event will
be given to the related constraints of the modified variable, to check consistency and propagate this new
information to the other variables.
Depending on the type of domain, events existing in Choco are:
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Event
description
Integer variable IntDomainVar
REMVAL
Remove a single value from the domain.
INCINF
Increase the lower bound of the domain.
DECSUP
Decrease the upper bound of the domain.
INSTINT Instantiate the domain, i.e. reduce it to a single value.
Set variable SetVar
REMENV
Remove a single value from the envelope domain
ADDKER
Add a single value to the kernel domain
INSTSET Instantiate the domain, i.e. set values to both kernel and envelope domain.
Real variable RealVar
INCINF
Increase the lower bound of the domain
DECSUP
Decrease the upper bound of the domain
An event given as a parameter to the engine is then pushed into a unique queue, waiting to be treated.
There are seven different queues where an event can be pushed, it depends on the priority of the event.
Beware, an event is not automatically pushed in a queue: if an other event based on the same variable
is already present in the queue, the two events are merged into one.
Due to promotion, variables’ events can be treated in a different order
than creation one!
Let X and Y, two integer variables with priorities of the same rank. During the propagation of a constraint, the lower bound of a integer variable
X is updated to 3 (this event is added to the queue Q), then Y is instantiated to 4 (this other event is added to the same queue Q) and finally
the upper bound of X is updated to 4 (the already pushed event on X
is updated with this new information). At that point, there are only 2
events in the queue Q: one on X and one on Y. As X and Y have the
same rank, events will be treated by creation order: event on X first ant
event on Y then. But the event on X contains 2 informations, one on the
lower bound modification and another on the upper bound modification.
So, although the instantiation of Y has been created first, the treatment
of the upper bound modification of X will be treated first.
Original event can be promoted : for example removing the last but one
value of an integer variable is promoted to instantiation.
Promotion are:
Original event can be promoted to
Integer variable IntDomainVar
REMVAL
INCINF or DECSUP or INSTINT
INCINF
DECSUP
INSTINT
INSTINT
INSTINT
REMENV
ADDKER
INSTSET
INSTSET
INSTSET
none
Set variable SetVar
none
Real variable RealVar
INCINF
none
DECSUP
none
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4.3
Define your own search strategy
Section Search strategy presented the default branching strategies available in Choco and showed how to
post them or to compose them as goals. In this section, we will start with a very simple and common way
to branch by choosing values for variables and specially how to define its own variable/value selection
strategy. We will then focus on more complex branching such as dichotomic or n-ary choices. Finally we
will show how to control the search space in more details with well known strategy such as LDS (Limited
discrepancy search).
Reminder : How does the search loop work?
For integer variables, the variable and value selection strategy objects are based on the following
interfaces:
• AbstractIntBranchingStrategy: abstract class for the branching strategy,
• VarSelector<V> : Interface for the variable selection (V extends Var),
• ValIterator<V> : Interface to describes an iteration scheme on the domain of a variable,
• ValSelector<V> : Interface for a value selection.
Concrete examples of these interfaces are respectively, AssignVar, MinDomain, IncreasingDomain,
MaxVal.
4.3.1
How to define your own Branching object
When you need a specific branching strategy that can’t be expressed with the ones already existing, you
can define your own concrete class of:
Default class to implement
definition
AbstractBinIntBranchingStrategy abstract class defining a binary tree search
AbstractLargeIntSConstraint
abstract class defining a n-ary tree search.
BranchingStrategy
IntBranching
AbstractBranchingStrategy
AbstractIntBranchingStrategy
AbstractBinIntBranchingStrategy
AbstractLargeIntBranchingStrategy
implementation
implementation
Figure 4.3: Branching strategy: interfaces and abstract classes.
We give here two examples of implementations of these classes, first for a binary branching, then for
a n-ary branching.
/**
* A class for branching schemes that consider two branches:
* - one assigning a value to an IntVar (X == v)
* - and the other forbidding this assignment (X != v)
*/
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public class AssignOrForbid extends AbstractBinIntBranchingStrategy {
protected VarSelector<IntDomainVar> varSelector;
protected ValSelector<IntDomainVar> valSelector;
public AssignOrForbid(VarSelector<IntDomainVar> varSelector,
ValSelector<IntDomainVar> valSelector) {
super();
this.valSelector = valSelector;
this.varSelector = varSelector;
}
/**
* Select the variable to constrained
*
* @return the branching object
*/
public Object selectBranchingObject() throws ContradictionException {
return varSelector.selectVar();
}
/**
* Select the value to assign, and set it in the decision object in parameter
* @param decision the next decision to apply
*/
@Override
public void setFirstBranch(final IntBranchingDecision decision) {
decision.setBranchingValue(valSelector.getBestVal(decision.getBranchingIntVar()));
}
/**
* Create and return the message to print, in case of strong verbosity
* @param decision current decision
* @return pretty print of the current decision
*/
@Override
public String getDecisionLogMessage(final IntBranchingDecision decision) {
return decision.getBranchingObject() + (decision.getBranchIndex() == 0 ? "==" : "=/=") +
decision.getBranchingValue();
}
/**
* Execution action based on the couple: {decision, branching index}.
* As <code>this</code> build a binary branching, there are only 2 branching indices:
* 0 -- assignment, the variable is instantiated to the value
* 1 -- forbidance, the value is removed from the domain of the variable
*
* @throws ContradictionException if the decision leads to an incoherence
*/
@Override
public void goDownBranch(final IntBranchingDecision decision) throws ContradictionException
{
if (decision.getBranchIndex() == 0) {
decision.setIntVal();
} else {
decision.remIntVal();
}
}
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}
/**
* A class for branching schemes that consider n branches:
* - assigning a value v_i to an variable (X == v_i)
*/
public class Assign extends AbstractLargeIntBranchingStrategy {
protected final VarSelector<IntDomainVar> varSelector;
protected ValIterator<IntDomainVar> valIterator;
public Assign(VarSelector<IntDomainVar> varSelector, ValIterator<IntDomainVar> valIterator)
{
this.varSelector = varSelector;
this.valIterator = valIterator;
}
/**
* Select the variable to constrained
*
* @return the branching object
*/
public Object selectBranchingObject() throws ContradictionException {
return varSelector.selectVar();
}
/**
* Select the first value to assign, and set it in the decision object in parameter
*
* @param decision the first decision to apply
*/
public void setFirstBranch(final IntBranchingDecision decision) {
decision.setBranchingValue(valIterator.getFirstVal(decision.getBranchingIntVar()));
}
/**
* Select the next value to assign, and set it in the decision object in parameter
*
* @param decision the next decision to apply
*/
public void setNextBranch(final IntBranchingDecision decision) {
decision.setBranchingValue(valIterator.getNextVal(decision.getBranchingIntVar(),
decision.getBranchingValue()));
}
/**
* Check wether there is still a value to assign
*
* @param decision the last decision applied
* @return <code>false</code> if there is still a branching to do
*/
public boolean finishedBranching(final IntBranchingDecision decision) {
return !valIterator.hasNextVal(decision.getBranchingIntVar(), decision.getBranchingValue
());
}
/**
* Apply the computed decision building the i^th branch.
* --> assignment: the variable is instantiated to the value
*
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*
* @param decision the decision to apply.
* @throws ContradictionException if the decision leads to an incoherence
*/
@Override
public void goDownBranch(final IntBranchingDecision decision) throws ContradictionException
{
decision.setIntVal();
}
/**
* Reconsider the computed decision, destroying the i^th branch
* --> forbiddance: the value is removed from the domain of the variable
*
* @param decision the decision that has been set at the father choice point
* @throws ContradictionException if the non-decision leads to an incoherence
*/
@Override
public void goUpBranch(final IntBranchingDecision decision) throws ContradictionException {
decision.remIntVal();
}
/**
* Create and return the message to print, in case of strong verbosity
* @param decision current decision
* @return pretty print of the current decision
*/
@Override
public String getDecisionLogMessage(IntBranchingDecision decision) {
return decision.getBranchingObject() + "==" + decision.getBranchingValue();
}
}
4.3.2
Define your own variable selection
VarSelector<V extends Var>
AbstractIntVarSelector
implementation
implementation
Figure 4.4: Variable selector: interface
You may extend this small library of branching schemes and heuristics by defining your own concrete
classes of AbstractIntVarSelector. We give here an example of an VarSelector<IntDomainVar> with the
implementation of a static variable ordering :
/**
* A variable selector selecting the first non instantiated variable according to a given
static order
*/
public class StaticVarOrder extends AbstractIntVarSelector {
private final IStateInt last;
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public StaticVarOrder(Solver solver) {
super(solver);
this.last = solver.getEnvironment().makeInt(0);
}
public StaticVarOrder(Solver solver, IntDomainVar[] vars) {
super(solver, vars);
this.last = solver.getEnvironment().makeInt(0);
}
/**
* Select the next uninstantiated variable, according to the define policy: input order
* @return the selected variable if exists, <code>null</code> otherwise
*/
public IntDomainVar selectVar() {
for (int i = last.get(); i < vars.length; i++) {
if (!vars[i].isInstantiated()) {
last.set(i);
return vars[i];
}
}
return null;
}
}
Notice on this example that you only need to implement method selectVar() which belongs to the
contract of VarSelector. This method should return a non instantiated variable or null. Once the
branching is finished, the next branching (if one exists) is taken by the search algorithm to continue the
search, otherwise, the search stops as all variable are instantiated. To avoid the loop over the variables
of the branching, a backtrackable integer (IStateInt) could be used to remember the last instantiated
variable and to directly select the next one in the table. Notice that backtrackable structures could be
used in any of the code presented in this chapter to speedup the computation of dynamic choices.
VarSelector<V extends Var>
AbstractRealVarSelector
AbstractSetVarSelector
implementation
implementation
AbstractIntVarSelector
implementation
TiedIntVarSelector
HeuristicIntVarSelector
IntHeuristicIntVarSelector
DoubleHeuristicIntVarSelector
implementation
implementation
Figure 4.5: Variable selector: interface and abstract classes
If you need an integer variable selector that can be used as a parameter of LexIntVarSelector, it should
extend IntHeuristicIntVarSelector or DoubleHeuristicIntVarSelector. These two abstract classes only
require to implement one method : getHeuristic(IntDomainVar v) which computes and returns a criterion
(int or double). The criteria are used in a master class to select the smallest crtierion’s variable
We give here an other example of an VarSelector<IntDomainVar>, this one extends IntHeuristicIntVarSelector
and choose the variable with the smallest domain :
public class MinDomain extends IntHeuristicIntVarSelector {
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public MinDomain(Solver solver) {
super(solver);
}
public MinDomain(Solver solver, IntDomainVar[] vs) {
super(solver, vs);
}
/**
* Compute the criterion, according to the define policy: smallest domain size
* @return the selected variable if exists, <code>null</code> otherwise
*/
@Override
public int getHeuristic(IntDomainVar v) {
return v.getDomainSize();
}
}
You can add your variable selector as a part of a search strategy, using solver.addGoal().
4.3.3
Define your own value selection
You may also define your own concrete classes of ValIterator or ValSelector.
Value selector
ValSelector<V extends Var>
implementation
Figure 4.6: Value selector: interface
We give here an example of an IntValSelector with the implementation of a minimum value selecting:
public class MinVal implements ValSelector<IntDomainVar> {
/**
* selecting the lowest value in the domain
*
* @param x the variable under consideration
* @return what seems the most interesting value for branching
*/
public int getBestVal(IntDomainVar x) {
return x.getInf();
}
}
Only getBestVal() method must be implemented, returning the best value in the domain according to
the heuristic.
You can add your value selector as a part of a search strategy, using solver.addGoal().
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Using a value selector with bounded domain variable is strongly inadvised,
except if it pick up bounds value. If the value selector pick up a value
that is not a bound, when it goes up in the tree search, that value could
be not removed and picked twice (or more)!
Values iterator
ValIterator<V extends Var>
implementation
Figure 4.7: Value iterator: interface
We give here an example of an ValIterator with the implementation of an increasing domain iterator:
public class IncreasingDomain implements ValIterator<IntDomainVar> {
/**
* testing whether more branches can be considered after branch i, on the alternative
associated to variable x
*
* @param x the variable under scrutiny
* @param i the index of the last branch explored
* @return true if more branches can be expanded after branch i
*/
public boolean hasNextVal(IntDomainVar x, int i) {
return (i < x.getSup());
}
/**
* Accessing the index of the first branch for variable x
*
* @param x the variable under scrutiny
* @return the index of the first branch (such as the first value to be assigned to the
variable)
*/
public int getFirstVal(IntDomainVar x) {
return x.getInf();
}
/**
* generates the index of the next branch after branch i, on the alternative associated to
variable x
*
* @param x the variable under scrutiny
* @param i the index of the last branch explored
* @return the index of the next branch to be expanded after branch i
*/
public int getNextVal(IntDomainVar x, int i) {
return x.getNextDomainValue(i);
}
}
You can add your value iterator as a part of a search strategy, using solver.addGoal().
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4.4
Define your own limit search space
To define your own limits/statistics (notice that a limit object can be used only to get statistics about the
search), you can create a limit object by extending the AbstractGlobalSearchLimit class or implementing
directly the interface IGlobalSearchLimit. Limits are managed at each node of the tree search and
are updated each time a node is open or closed. Notice that limits are therefore time consuming.
Implementing its own limit need only to specify to the following interface :
/**
* The interface of objects limiting the global search exploration
*/
public interface GlobalSearchLimit {
/**
* resets the limit (the counter run from now on)
* @param first true for the very first initialization, false for subsequent ones
*/
public void reset(boolean first);
/**
* notify the limit object whenever a new node is created in the search tree
* @param solver the controller of the search exploration, managing the limit
* @return true if the limit accepts the creation of the new node, false otherwise
*/
public boolean newNode(AbstractGlobalSearchSolver solver);
/**
* notify the limit object whenever the search closes a node in the search tree
* @param solver the controller of the search exploration, managing the limit
* @return true if the limit accepts the death of the new node, false otherwise
*/
public boolean endNode(AbstractGlobalSearchSolver solver);
}
Look at the following example to see a concrete implementation of the previous interface. We define
here a limit on the depth of the search (which is not found by default in choco). The getWorldIndex()
is used to get the current world, i.e the current depth of the search or the number of choices which have
been done from baseWorld.
public class DepthLimit extends AbstractGlobalSearchLimit {
public DepthLimit(AbstractGlobalSearchSolver theSolver, int theLimit) {
super(theSolver,theLimit);
unit = "deep";
}
public boolean newNode(AbstractGlobalSearchSolver solver) {
nb = Math.max(nb, this.getProblem().getWorldIndex()
this.getProblem().getSolver().getSearchSolver().baseWorld);
return (nb < nbMax);
}
public boolean endNode(AbstractGlobalSearchSolver solver) {
return true;
}
public void reset(boolean first) {
if (first) {
nbTot = 0;
} else {
nbTot = Math.max(nbTot, nb);
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}
nb = 0;
}
Once you have implemented your own limit, you need to tell the search solver to take it into account.
Instead of using a call to the solve() method, you have to create the search solver by yourself and add
the limit to its limits list such as in the following code :
Solver s = new CPSolver();
s.read(model);
s.setFirstSolution(true);
s.generateSearchStrategy();
s.getSearchStrategy().limits.add(new DepthLimit(s.getSearchStrategy(),10));
s.launch();
4.5
Define your own constraint
This section describes how to add you own constraint, with specific propagation algorithms. Note that
this section is only useful in case you want to express a constraint for which the basic propagation
algorithms (using tables of tuples, or boolean predicates) are not efficient enough to propagate the
constraint.
The general process consists in defining a new constraint class and implementing the various propagation methods. We recommend the user to follow the examples of existing constraint classes (for instance,
such as GreaterOrEqualXYC for a binary inequality)
4.5.1
The constraint hierarchy
Each new constraint must be represented by an object implementing the SConstraint interface (S for
solver constraint). To help the user defining new constraint classes, several abstract classes defining
SConstraint have been implemented. These abstract classes provide the user with a management of
the constraint network and the propagation engineering. They should be used as much as possible.
For constraints on integer variables, the easiest way to implement your own constraint is to inherit
from one of the following classes, depending of the number of solver integer variables (IntDomainVar)
involved:
Default class to implement
number of solver integer variables
AbstractUnIntSConstraint
one variable
AbstractBinIntSConstraint
two variables
AbstractTernIntSConstraint
three variables
AbstractLargeIntSConstraint any number of variables.
Constraints over integers must implement the following methods (grouped in the IntSConstraint interface):
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Method to implement
pretty()
propagate()
awake()
awakeOnInst(int x)
awakeOnBounds(int x)
awakeOnRemovals(int x, IntIterator v)
description
Returns a pretty print of the constraint
The main propagation method (propagation from scratch).
Propagating the constraint until local consistency is reached.
Propagating the constraint for the very first time until local
consistency is reached. The awake is meant to initialize the
data structures contrary to the propagate. Specially, it is important to avoid initializing the data structures in the constructor.
Default propagation on instantiation: full constraint repropagation.
Default propagation on improved bounds: propagation on domain revision.
Default propagation on mutliple values removal: propagation
on domain revision. The iterator allow to iterate over the values
that have been removed.
Methods awakeOnBounds and awakeOnRemovals can be replaced by more fine grained methods:
awakeOnInf(int x)
Default propagation on improved lower bound: propagation on
domain revision.
awakeOnSup(int x)
Default propagation on improved upper bound: propagation
on domain revision.
awakeOnRem(int x, int v)
Default propagation on one value removal: propagation on domain revision.
To use the constraint in expressions or reification, the following minimum API is mandatory:
isSatisfied(int[] x)
Tests if the constraint is satisfied when the variables are instantiated.
isEntailed()
Checks if the constraint must be checked or must fail. It returns
true if the constraint is known to be satisfied whatever happend
on the variable from now on, false if it is violated.
opposite()
It returns an AbstractSConstraint that is the opposite of the
current constraint.
In the same way, a set constraint can inherit from AbstractUnSetSConstraint, AbstractBinSetSConstraint,
AbstractTernSetSConstraint or AbstractLargeSetSConstraint.
A real constraint can inherit from AbstractUnRealSConstraint, AbstractBinRealSConstraint
or AbstractLargeRealSConstraint.
A mixed constraint between set and integer variables can inherit from AbstractBinSetIntSConstraint
or AbstractLargeSetIntSConstraint.
A simple way to implement its own constraint is to:
• create an empty constraint with only propagate()
method implemented and every awakeOnXxx() ones set to
this.constAwake(false);
• when the propagation filter is sure, separate it into the
awakeOnXxx() methods in order to have finer granularity
• finally, if necessary, use backtrackables objects to improve the efficient of your constraint
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Interact with variables
One of the constraint function is to remove forbidden values from domain variable (filtering). Variables
provide services to allow constraint to interact with their domain.
IntDomainVar
boolean removeVal(int x, final SConstraint cause, final boolean forceAwake)
Update the domain of the integer variable by removing x from the domain. cause is the constraint
at the origin of the event, forceAwake indicates wether or not the cause constraint must be informed
of this event. The result of such call can be true, the value has been removed without any trouble,
false the value was not present in the domain. A ContradictionException is thrown if it empties
the domain of the variable
boolean removeInterval(int a, int b, final SConstraint cause, final boolean forceAwake)
Update the domain of the integer variable by removing all values contained in the interval [a, b]
from the domain. cause is the constraint at the origin of the event, forceAwake indicates wether or
not the cause constraint must be informed of this event. The result of such call can be true, the
values has been removed without any trouble, false if the intersection of the current domain and
[a, b] was empty. A ContradictionException is thrown if it empties the domain of the variable.
boolean updateInf(int x, final SConstraint cause, final boolean forceAwake)
Update the domain of the integer variable by removing all values strictly below x from the domain.
cause is the constraint at the origin of the event, forceAwake indicates wether or not the cause
constraint must be informed of this event. The result of such call can be true, the lower bound has
been updated without any trouble, false the new lower bound was smaller or equal to the actual
one. A ContradictionException is thrown if it empties the domain of the variable.
boolean updateSup(int x, final SConstraint cause, final boolean forceAwake)
Update the domain of the integer variable by removing all values strictly above x from the domain.
cause is the constraint at the origin of the event, forceAwake indicates wether or not the cause
constraint must be informed of this event. The result of such call can be true, the upper bound
has been updated without any trouble, false the new upper bound was greater or equal to the
actual one. A ContradictionException is thrown if it empties the domain of the variable.
boolean instantiate(int x, final SConstraint cause, final boolean forceAwake)
Update the domain of the integer variable by removing all values but x from the domain. cause is
the constraint at the origin of the event, forceAwake indicates wether or not the cause constraint
must be informed of this event. The result of such call can be true, the domain has been reduced
to a singleton without any trouble, false the domain was already instantiated to the same value
. A ContradictionException is thrown if x is out of the domain or if the domain was already
instantiated to another value.
SetVar
boolean remFromEnveloppe(int x, final SConstraint cause, final boolean forceAwake)
Update the domain of the set variable by removing x from the envelope’s domain. cause is the
constraint at the origin of the event, forceAwake indicates wether or not the cause constraint must
be informed of this event. The result of such call can be true, the value has been removed without
any trouble, false the value was not present in the envelope. A ContradictionException is thrown
if x is present in the kernel’s domain.
boolean addToKernel(int x, final SConstraint cause, final boolean forceAwake)
Update the domain of the set variable by adding x into the kernel’s domain. cause is the constraint
at the origin of the event, forceAwake indicates wether or not the cause constraint must be informed
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of this event. The result of such call can be true, the value has been added without any trouble,
false the value was already present in the kernel. A ContradictionException is thrown if x is not
present in the envelope’s domain.
boolean instantiate(int[] xs, final SConstraint cause, final boolean forceAwake)
Update the domain of the set variable by removing every values but those in xs from the envelope’s
domain and by adding every values of xs into the kernel’s domain. A set variable is known as
instantiated when E ∩ K 6= ∅ and E 4 K = ∅. cause is the constraint at the origin of the event,
forceAwake indicates wether or not the cause constraint must be informed of this event. The result
of such call can be true, the envelope or the kernel have been updated without trouble, false the
envelope and the kernel were already equal to xs . A ContradictionException is thrown if the at
least one value from xs is not present in the kernel’s domain or in the envelope’s domain.
RealVar
void intersect(RealInterval interval)
Update the domain of the real variable by intersecting the domain with interval (define at least by
two doubles, known as lower and upper bound). A ContradictionException is thrown the resulting
interval is incoherent (the new lower bound is greater than the new upper bound).
How do I add my constraint to the Model ?
Adding your constraint to the model requires you to definite a specific constraint manager (that can be
a inner class of your Constraint). This manager need to implement:
makeConstraint(Solver s, Variable[] vars, Object params, HashSet<String> options)
This method allows the Solver to create an instance of your constraint, with your parameters and Solver
objects.
If you create your constraint manager as an inner class, you must declare
this class as public and static. If you don’t, the solver can’t instantiate
your manager.
Once this manager has been implemented, you simply add your constraint to the model using the
addConstraint() API with a ComponentConstraint object:
model.addConstraint( new ComponentConstraint(MyConstraintManager.class, params, vars) );
// OR
model.addConstraint( new ComponentConstraint("package.of.MyConstraint", params, vars) );
Where params is whatever you want (Object[], int, String,...) and vars is an array of Model Variables (or more specific) objects.
4.5.2
Example: implement and add the IsOdd constraint
One creates the constraint by implementing the AbstractUnIntSConstraint (one integer variable) class:
public class IsOdd extends AbstractUnIntSConstraint {
@Override
public int getFilteredEventMask(int idx) {
return IntVarEvent.INSTINT_MASK;
}
public IsOdd(IntDomainVar v0) {
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super(v0);
}
/**
* Default initial propagation: full constraint re-propagation.
*/
public void awake() throws ContradictionException {
DisposableIntIterator it = v0.getDomain().getIterator();
try{
while(it.hasNext()){
int val = it.next();
if(val%2==0){
v0.removeVal(val, this, false);
}
}
}finally {
it.dispose();
}
}
/**
* <i>Propagation:</i>
* Propagating the constraint until local consistency is reached.
*
* @throws ContradictionException
* contradiction exception
*/
public void propagate() throws ContradictionException {
if(v0.isInstantiated()){
if(v0.getVal()%2==0){
fail();
}
}
}
}
To add the constraint to the model, one creates the following class (or inner class):
public class IsOddManager extends IntConstraintManager {
public SConstraint makeConstraint(Solver solver, IntegerVariable[] variables, Object
parameters, List<String> options) {
if (solver instanceof CPSolver) {
return new IsOdd(solver.getVar(variables[0]));
}
return null;
}
}
It calls the constructor of the constraint, with every vars, params and options needed.
Then, the constraint can be added to a model as follows:
// Creation of the model
Model m = new CPModel();
// Declaration of the variable
IntegerVariable aVar = Choco.makeIntVar("a_variable", 0, 10);
// Adding the constraint to the model, 1st solution:
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m.addConstraint(new ComponentConstraint(IsOddManager.class, null, new IntegerVariable[]{aVar
}));
// OR 2nd solution:
m.addConstraint(new ComponentConstraint("myPackage.Constraint.IsOddManager", null, new
IntegerVariable[]{aVar}));
Solver s = new CPSolver();
s.read(m);
s.solve();
And that’s it!!
4.5.3
Example of an empty constraint
public class ConstraintPattern extends AbstractLargeIntSConstraint {
public ConstraintPattern(IntDomainVar[] vars) {
super(vars);
}
/**
* pretty print. The String is not constant and may depend on the context.
* @return a readable string representation of the object
*/
public String pretty() {
return null;
}
/**
* check whether the tuple satisfies the constraint
* @param tuple values
* @return true if satisfied
*/
public boolean isSatisfied(int[] tuple) {
return false;
}
/**
* propagate until local consistency is reached
*/
public void propagate() throws ContradictionException {
// elementary method to implement
}
/**
* propagate for the very first time until local consistency is reached.
*/
public void awake() throws ContradictionException {
constAwake(false); // change if necessary
}
/**
* default propagation on instantiation: full constraint re-propagation
* @param var index of the variable to reduce
*/
public void awakeOnInst(int var) throws ContradictionException {
constAwake(false); // change if necessary
}
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4.6. DEFINE YOUR OWN OPERATOR
/**
* default propagation on improved lower bound: propagation on domain revision
* @param var index of the variable to reduce
*/
public void awakeOnInf(int var) throws ContradictionException {
constAwake(false); // change if necessary
}
/**
* default propagation on improved upper bound: propagation on domain revision
* @param var index of the variable to reduce
*/
public void awakeOnSup(int var) throws ContradictionException {
constAwake(false); // change if necessary
}
/**
* default propagation on improve bounds: propagation on domain revision
* @param var index of the variable to reduce
*/
public void awakeOnBounds(int var) throws ContradictionException {
constAwake(false); // change if necessary
}
/**
* default propagation on one value removal: propagation on domain revision
* @param var index of the variable to reduce
* @param val the removed value
*/
public void awakeOnRem(int var, int val) throws ContradictionException {
constAwake(false); // change if necessary
}
/**
* default propagation on one value removal: propagation on domain revision
* @param var index of the variable to reduce
* @param delta iterator over remove values
*/
public void awakeOnRemovals(int var, IntIterator delta) throws ContradictionException {
constAwake(false); // change if necessary
}
}
The first step to create a constraint in Choco is to implement all awakeOn... methods with
constAwake(false) and to put your propagation algorithm in the propagate() method.
A constraint can choose not to react to fine grained events such as the removal of a value of a given
variable but instead delay its propagation at the end of the fix point reached by “fine grained events” and
fast constraints that deal with them incrementally (that’s the purpose of the constraints events queue).
To do that, you can use constAwake(false) that tells the solver that you want this constraint to
be called only once the variables events queue is empty. This is done so that heavy propagators can
delay their action after the fast one to avoid doing a heavy processing at each single little modification
of domains.
4.6
Define your own operator
There are 2 types of operators: boolean and arithmetic. These operators are based on integer variable
and/or integer constants. Let’s take 2 examples plus and eq.
The operator plus is an arithmetic one (see PlusNode.java class for details) that computes the sum
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CHAPTER 4. ADVANCED USES OF CHOCO
of two variables (and/or constants). It extends INode (it can be part of an expression object) and
implements ArithmNode (it can be evaluated). So, required services are:
• a constructor. The type of operator should be defined using CUSTOM.
public PlusNode(INode[] subt) {
super(subt, NodeType.CUSTOM);
}
• pretty() : a pretty print of the operator
public String pretty() {
return "("+subtrees[0].pretty()+" + "+subtrees[1].pretty()+")";
}
• eval(int[] tuple) : evaluation of the operator with the given tuple. An arithmetic evaluation of
the subtrees is done (based on the given tuple) to compute the sub expression before evaluating
the current operator plus. This allows tree-like representation of an expression.
public int eval(int[] tuple) {
return ((ArithmNode) subtrees[0]).eval(tuple) + ((ArithmNode) subtrees[1]).eval(tuple
);
}
The operator eq is a boolean one (see EqNode.java class for more details) that checks if two variables
(and/or constants) are equal. It extends AbstractBooleanNode (that can be checked). So required services
are:
• a constructor
public EqNode(INode[] subt) {
super(subt, NodeType.CUSTOM);
}
• pretty() : a pretty print of the operator
public String pretty() {
return "("+subtrees[0].pretty()+"="+subtrees[1].pretty()+")";
}
• eval(int[] tuple): an expression checker, based on the given tuple. An arithmetic evaluation of
the subtrees is done (based on the given tuple) in order to check the operator eq.
public boolean checkTuple(int[] tuple) {
return ((ArithmNode) subtrees[0]).eval(tuple)
== ((ArithmNode) subtrees[1]).eval(tuple);
}
• extractConstraint(Solver s) : extracts the corresponding constraint in intension constraint without reification.
public SConstraint extractConstraint(Solver s) {
IntDomainVar v1 = subtrees[0].extractResult(s);
IntDomainVar v2 = subtrees[1].extractResult(s);
return s.eq(v1,v2);
}
Now let’s see how to use this operator in a Model.
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4.7. DEFINE YOUR OWN VARIABLE
To do that, create your own manager implementing ExpressionManager, which makes the link between
the model and the solver.
Then to use your operator in your model, you can define a static method to simplify the calls.
Let’s sum up in a short example based on the plus operator.
The manager would be:
public class PlusManager implements ExpressionManager {
@Override
public INode makeNode(Solver solver, Constraint[] cstrs, Variable[] vars) {
if(solver instanceof CPSolver){
CPSolver s = (CPSolver)solver;
if(vars.length == 1){
INode[] nodes = new INode[vars.length];
for(int v = 0; v < vars.length; v++){
nodes[v] = vars[v].getExpressionManager().makeNode(s, vars[v].getConstraints
(), vars[v].getVariables());
}
return new PlusNode(nodes);
}
}
throw new ModelException("Could not found a node manager in " + this.getClass() + " !");
}
}
A main class would be:
public class Sandbox {
public static void main(String[] args) {
model1();
}
public static IntegerExpressionVariable plus(IntegerVariable x){
return new IntegerExpressionVariable(null, "package.of.PlusManager", x);
}
private static void model1() {
Model m = new CPModel();
IntegerVariable x = Choco.makeIntVar("x", 0, 5);
IntegerVariable y = Choco.makeIntVar("y", 4, 8);
IntegerVariable z = Choco.makeIntVar("z", 0, 10);
// declare an expression using my operator
IntegerExpressionVariable xx = plus(x,y);
// and use it in common constraint
m.addConstraint(Choco.eq(z, xx));
Solver s = new CPSolver();
s.read(m);
ChocoLogging.toSolution();
s.solveAll();
}
}
Keep in mind that you can not define operators for set and real.
4.7
Define your own variable
to complete
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4.8
Model and Solver detectors
Sometimes, on automatic code generation or during benchmarking, it could be useful to apply generic
rules to analyze a Model and detect lacks of modeling and apply patchs. With Choco, this is possible
using ModelDetectorFactory and PreProcessCPSolver.
4.8.1
Model detector
The analysis of a Model is done using the ModelDetectorFactory. First one declares the list of rules to
apply, then they are applied to a specific model. This is done by using the follwing API:
ModelDetectorFactory.run(CPModel model, AbstractDetector... detectors)
An AbstractDetector object describes the pattern to detect within the model and rules to apply. Applying
a rule is commonly to refactor a given model, by adding or removing variables and constraints.
Calling ModelDetectorFactory.run(..) will produce a modified copy of
the current model. Very few AbstractDetector just analyze the model,
without any side effects.
Detectors
AbstractDetector analysis(CPModel m) Analyze the model m, and print out messages about gen-
eral statistics: very large domain variables, inappropriate domain type variables, free variables
(variables not involved in any constraints), etc.
AbstractDetector intVarEqDet(CPModel m)
AbstractDetector taskVarEqDet(CPModel m)
AbstractDetector cliqueDetector(CPModel m, boolean breakSymetries)
AbstractDetector disjFromCumulDetector(CPModel m)
AbstractDetector precFromImpliedDetector(CPModel m, DisjunctiveModel disjMod)
AbstractDetector precFromReifiedDetector(CPModel m, DisjunctiveModel disjMod)
AbstractDetector precFromTimeWindowDetector(CPModel m, DisjunctiveModel disjMod)
AbstractDetector precFromDisjointDetector(CPModel m, DisjunctiveModel disjMod)
AbstractDetector disjointDetector(CPModel m, DisjunctiveModel disjMod)
AbstractDetector disjointFromDisjDetector(CPModel m, DisjunctiveModel disjMod)
AbstractDetector disjointFromCumulDetector(CPModel m, DisjunctiveModel disjMod)
AbstractDetector rmDisjDetector(final CPModel m)
AbstractDetector[] disjunctiveModelDetectors(CPModel m, DisjunctiveModel disjMod)
AbstractDetector[] schedulingModelDetectors(final CPModel m, DisjunctiveModel disjMod)
Analysing a Model can be time consuming. It should be used carefully.
to complete
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4.9. LOGGING STATEMENTS
4.8.2
Preprocess solver
One may want to keep the Model unchanged but apply detection directly on the Solver. This can be
done during the reading step: the model is analyses on the fly and the rules are applied directly on the
solver.
To do that, simply replaced the CPSolver declaration by:
Solver solver = new PreProcessCPSolver(); // new CPSolver();
The rules that will be applied can be selected by updating the PreProcessConfiguration file. to
complete
4.9
4.9.1
Logging statements
Define your own logger.
ChocoLogging.makeUserLogger(String suffix);
4.9.2
How to write logging statements ?
• Critical Loggers are provided to display error or warning. Displaying too much message really
impacts the performances.
• Check the logging level before creating arrays or strings.
• Avoid multiple calls to Logger functions. Prefer to build a StringBuilder then call the Logger
function.
• Use the Logger.log function instead of building string in Logger.info().
4.9.3
Handlers.
Logs are displayed on System.out but warnings and severe messages are also displayed on System.err.
ChocoLogging.java also provides utility functions to easily change handlers:
• Functions set...Handler remove current handlers and replace them by a new handler.
• Functions add...Handler add new handlers but do not touch existing handlers.
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Chapter 5
Choco and CP-Viz
This chapter presents how to produce XML files for the CP-Viz system. CP-VIz is an open source
visualization platform for CP. It provides multiple views to show the search tree, and the state of
constraints and variables for a post- mortem analysis of a constraint program. It is system independent,
uses XML format to exchange information between solvers and visualization tools.
Find out more information about CP-Viz in [Simonis et al., 2010]: A Generic Visualization Platform
for CP (CP2010 paper and presentation).
Although visualization is really helpful, we do want the production of such files being easily unpluggable and not too much intrusive. That is why we have chosen Aspect-oriented programming, more
precisely the AspectJ library, and a small footprint logging system: LOGBack to implement this fonctionnality in Choco.
5.1
Three step tutorial
Producing CP-Viz traces in your Choco program can be done in three steps:
• first change the choco-solver JAR for the choco-solver-cpviz one in your classpath (it contains
choco-solver, the tracer tools and the dependencies). This JAR file does not require any additionnal JARs in the classpath;
• then, add the Visualization object and options and the optionnal Visualizer objects (see below);
• and go to your.cpviz.dir/viz/bin, and run the following command:
java ie.ucc.cccc.viz.Viz configuration.xml tree.xml visualization.xml
.
The two first steps produce XML files required by CP-Viz as parameter. The third step generates
SVG and IDX files (see the CP-Viz documentation for more information).
Don’t mix up :
• XML outputs, produced by Choco
• and files generated by CP-Viz from XML outputs.
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CHAPTER 5. CHOCO AND CP-VIZ
5.2
Visualization and Visualizers
5.2.1
Visualization
The Visualization object is mandatory to produce CP-Viz files. It creates, using the logging system, the
2 or 3 required files: configuration-{pbname}.xml, tree-{pbname}.xml and visualization-{pbname}.xml,
where {pbname} is the problem name and can be specified at the Visualization creation. These files are
created at the project root directory (this can be changed by overriding logback.xml file, see LOGBack).
Visualization has 2 constructors:
• Visualization(String pbname, Solver solver, String dir): create a new instance of Visualization
, where :
– pbname is the name of the problem (and the suffix of the XML files),
– solver is the Solver object used to solve the problem,
– dir is the path used by CP-Viz to generate SVG files and IDX file (see the CP-Viz documentation for more information). This must be an existing directory, otherwise CP-Viz can not
create anything!
• Visualization(Solver solver, String dir): identical to the previous one, except the pbname is
generated using new Random().nextLong().
and provide 6 services:
• void createTree(String type, String display, String repeat, int width, int height): declare
the tree search visualization, where :
–
–
–
–
–
type should take its value in {"layout", "graph", "value"}
display should take its value in {"compact", "expanded"}
repeat should take its value in {"all", "final", "i", "-i"}
width is the width of SVG canvas in screen pixels
height is the height of SVG canvas in screen pixels.
These parameters are used by CP-Viz to create SVG files. They can be modified afterwards in
the configuration-{pbname}.xml file. The generated file records informations about root node, try
nodes, fail nodes and success nodes of the tree search. Constants can also be found in CPVizConstant
class.
• void createTree(): default tree search declaration, with type="layout", display="compact", repeat
="all", width= 500, height= 500.
• void createViz(String type, String display, String repeat, int width, int height): declare the
constraint and variable visualizer container, where :
–
–
–
–
–
type should take its value in {"layout"}
display should take its value in {"compact", "expanded"}
repeat should take its value in {"all", "final", "i", "-i"}
width is the width of SVG canvas in screen pixels
height is the height of SVG canvas in screen pixels.
These parameters are used by CP-Viz to create SVG files. They can be modified afterwards in the
configuration-{pbname}.xml file. Constants can also be found in CPVizConstant class.
• void createViz(): default constraint and variable visualizers container declaration, with type="
layout", display="compact", repeat="all", width= 500, height= 500.
• void addVisualizer(Visualizer visualizer): add a visualizer to the constraint and variable visualizers container. See the Visualizers section for more details.
• void close(): safely close the XML files after the resolution – required.
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5.3. EXAMPLE OF CODE INTEGRATION
5.2.2
Visualizers
A Visualizer is a specific object containing informations about the way some variables should be represented in CP-Viz. They describe the state of a set of variables all tree search long.
Each visalizer is a specific extension of the abstract Visualizer class. This class declares common
parameters of visualizers and provides the following services:
• void setXY(int x, int y) : sets the coordinates of the visualizer in the visualizer window (default
is [0,0])
• void setGroup(String group): defines the group name of the visualizer
• void setMinMax(int min, int max): sets the expected minimal and maximal value of any of the
domains
Although there are no real differences between them, we distinguish two types of visualizers: variableoriented visualizers and constraint-oriented visualizers.
• Variable-oriented visualizers : BinaryMatrix, BinaryVector, DomainMatrix,
Vector, VectorSize, VectorWaterfall,
• Constraint-oriented visualizers : AllDifferent, AllDifferentMatrix,
BinPacking, BooleanChanneling, Cumulative, Element, Gcc, Inverse, LexLe,
LexLt,
Note that constraint-oriented visualizers are not based on any Constraint object but just Var object
(Solver objects). This gives more freedom in visualizer declaration (for example, on reformulations).
5.3
Example of code integration
Let’s see a visualization integration in the well-known SendMoreMoney problem:
Model model;
IntegerVariable S, E, N, D, M, O, R, Y;
IntegerVariable[] SEND, MORE, MONEY;
model = new CPModel();
S = makeIntVar("S", 0, 9);
E = makeIntVar("E", 0, 9);
N = makeIntVar("N", 0, 9);
D = makeIntVar("D", 0, 9);
M = makeIntVar("M", 0, 9);
O = makeIntVar("0", 0, 9);
R = makeIntVar("R", 0, 9);
Y = makeIntVar("Y", 0, 9);
SEND = new IntegerVariable[]{S, E, N, D};
MORE = new IntegerVariable[]{M, O, R, E};
MONEY = new IntegerVariable[]{M, O, N, E, Y};
model.addConstraints(neq(S, 0), neq(M, 0));
model.addConstraint(allDifferent(S, E, N, D, M, O, R, Y));
model.addConstraints(
eq(plus(scalar(new int[]{1000, 100, 10, 1}, SEND),
scalar(new int[]{1000, 100, 10, 1}, MORE)),
scalar(new int[]{10000, 1000, 100, 10, 1}, MONEY))
);
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Solver solver = new CPSolver();
solver.read(model);
//-------> Visualization declaration starts here <-------//
// create a new instance of Visualization
Visualization visu = new Visualization("SendMoreMoney", solver, "path/to/out");
visu.createTree(); // declare tree search visualization
visu.createViz(); // declare visualizers container
// create a new Vector visualizer
Vector visualizer = new Vector(solver.getVar(S, E, N, D, M, O, R, Y), "expanded", 0, 0, 8, 10,
"SENDMORY", 0, 9);
// add the vector to the visualizers container
visu.addVisualizer(visualizer);
// launch the resolution of the problem
solver.solve();
// close the visualization
visu.close();
//-------> Visualization declaration ends here <-------//
SendMoreMoney problem
The execution of this program results in 3 files:
<configuration version="1.0" directory="/path/to/out" >
<tool show="tree" type="layout" display="compact" repeat="all" width="500" height="500"
fileroot="tree-SendMoreMoney" />
<tool show="viz" type="layout" display="compact" repeat="all" width="500" height="500"
fileroot="visualization-SendMoreMoney" />
</configuration>
configuration-SendMoreMoney.xml
<?xml version="1.0" encoding="UTF-8"?>
<!-- choco-solver -->
<tree version="1.0" >
<root id="0" />
<fail id="1" parent="0" name="E" size="4" value="4" />
<try id="2" parent="0" name="E" size="1" value="5" />
<succ id="2" />
</tree>
tree-SendMoreMoney.xml
<?xml version="1.0" encoding="UTF-8"?>
<!-- choco-solver -->
<visualization version="1.0" >
<visualizer id="1" type="vector" display="expanded" width="8" height="10" x="0" y="0" group="
SENDMORY" min="0" max="9"/>
<state id="1" tree_node="-1" >
<visualizer_state id="1" >
<dvar index="1" domain="0 1 2 3 4 5 6 7 8 9 " />
<dvar index="2" domain="0 1 2 3 4 5 6 7 8 9 " />
<dvar index="3" domain="0 1 2 3 4 5 6 7 8 9 " />
<dvar index="4" domain="0 1 2 3 4 5 6 7 8 9 " />
<dvar index="5" domain="0 1 2 3 4 5 6 7 8 9 " />
<dvar index="6" domain="0 1 2 3 4 5 6 7 8 9 " />
<dvar index="7" domain="0 1 2 3 4 5 6 7 8 9 " />
<dvar index="8" domain="0 1 2 3 4 5 6 7 8 9 " />
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5.3. EXAMPLE OF CODE INTEGRATION
</visualizer_state>
</state>
<state id="2" tree_node="0" >
<visualizer_state id="1" >
<integer index="1" value="9" />
<dvar index="2" domain="4 5 6 7 " />
<dvar index="3" domain="5 6 7 8 " />
<dvar index="4" domain="2 3 4 5 6 7 8 " />
<integer index="5" value="1" />
<integer index="6" value="0" />
<dvar index="7" domain="2 3 4 5 6 7 8 " />
<dvar index="8" domain="2 3 4 5 6 7 8 " />
</visualizer_state>
</state>
<state id="3" tree_node="1" >
<visualizer_state id="1" >
<integer index="1" value="9" />
<dvar index="2" domain="4 5 6 7 " />
<dvar index="3" domain="5 6 7 8 " />
<dvar index="4" domain="2 3 4 5 6 7 8 " />
<integer index="5" value="1" />
<integer index="6" value="0" />
<dvar index="7" domain="2 3 4 5 6 7 8 " />
<dvar index="8" domain="2 3 4 5 6 7 8 " />
<failed index="2" group="SENDMORY" value="4"/>
</visualizer_state>
</state>
<state id="4" tree_node="2" >
<visualizer_state id="1" >
<integer index="1" value="9" />
<integer index="2" value="5" />
<integer index="3" value="6" />
<integer index="4" value="7" />
<integer index="5" value="1" />
<integer index="6" value="0" />
<integer index="7" value="8" />
<integer index="8" value="2" />
<focus index="2" group="SENDMORY" type="vector"/>
</visualizer_state>
</state>
</visualization>
visualization-SendMoreMoney.xml
With these 3 XML files, CP-Viz can produce the following outputs:
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CHAPTER 5. CHOCO AND CP-VIZ
E
4
4
1
5
Figure 5.1: Example of search tree output for SendMoreMoney resolution
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
9
Figure 5.2: Example of visualization output for SendMoreMoney resolution
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Part II
Elements of Choco
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Chapter 6
Variables (Model)
This section describes the three kinds of variables that can be used within a Choco Model, and an
object-variable.
6.1
Integer variables
IntegerVariable is a variable whose associated domain is made of integer values.
constructors:
Choco method
makeIntVar(String name, int lowB, int uppB, String... options)
makeIntVar(String name, int[] values, String... options)
makeIntVar(String name, List<Integer> values, String... options)
makeIntVar(String name, TIntArrayList values, String... options)
makeBooleanVar(String name, String... options)
makeIntVarArray(String name, int dim, int lowB, int uppB, String... options)
makeIntVarArray(String name, int dim, int[] values, String... options)
makeIntVarArray(String name, int dim, List<Integer> values, String...
options)
makeIntVarArray(String name, int dim, TIntArrayList values, String...
options)
makeBooleanVarArray(String name, int dim, String... options)
makeIntVarArray(String name, int dim1, int dim2, int lowB, int uppB, String
... options)
makeIntVarArray(String name, int dim1, int dim2, int[] values, String...
options)
makeIntVarArray(String name, int dim1, int dim2, List<Integer> values,
String... options)
makeIntVarArray(String name, int dim1, int dim2, TIntArrayList values,
String... options)
return type
IntegerVariable
IntegerVariable
IntegerVariable
IntegerVariable
IntegerVariable
IntegerVariable[]
IntegerVariable[]
IntegerVariable[]
IntegerVariable[]
IntegerVariable[]
IntegerVariable[][]
IntegerVariable[][]
IntegerVariable[][]
IntegerVariable[][]
options:
• no option : equivalent to option Options.V ENUM
• Options.V ENUM : to force Solver to create enumerated domain for the variable.
• Options.V BOUND : to force Solver to create bounded domain for the variable.
• Options.V LINK : to force Solver to create linked list domain for the variable.
• Options.V BTREE : to force Solver to create binary tree domain for the variable.
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CHAPTER 6. VARIABLES (MODEL)
• Options.V BLIST : to force Solver to create bipartite list domain for the variable.
• Options.V MAKEPSAN : declare the current variable as makespan.
• Options.V NO DECISION : to force variable to be removed from the pool of decisional variables.
• Options.V OBJECTIVE : to define the variable to be the one to optimize.
methods:
• removeVal(int val): remove value val from the domain of the current variable
A variable with {0, 1} domain is automatically considered as boolean domain.
Example:
IntegerVariable ivar1 = makeIntVar("ivar1", -10, 10);
IntegerVariable ivar2 = makeIntVar("ivar2", 0, 10000, Options.V_BOUND, Options.
V_NO_DECISION);
IntegerVariable bool = makeBooleanVar("bool");
Integer variables are illustrated on the n-Queens problem.
6.2
Real variables
RealVariable is a variable whose associated domain is made of real values. Only enumerated domain
is available for real variables.
Such domain are memory consuming. In order to minimize the memory use and to have the precision
you need, the model offers a way to set a precision (default value is 1.0e-6):
Solver m = new CPSolver();
m.setPrecision(0.01);
constructor:
Choco method
makeRealVar(String name, double lowB, double uppB, String... options)
return type
RealVariable
options:
• no option : no particular choice on decision or objective.
• Options.V NO DECISION : to force variable to be removed from the pool of decisional variables.
• Options.V OBJECTIVE : to define the variable to be the one to optimize.
Example:
RealVariable rvar1 = makeRealVar("rvar1", -10.0, 10.0);
RealVariable rvar2 = makeRealVar("rvar2", 0.0, 100.0, Options.V_NO_DECISION, Options.
V_OBJECTIVE);
Real variables are illustrated on the CycloHexan problem.
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6.3. SET VARIABLES
6.3
Set variables
SetVariable is high level modeling tool. It allows to represent variable whose values are sets. A
SetVariable on integer values between [1, n] has 2n values (every possible subsets of {1..n}). This makes
an exponential number of values and the domain is represented with two bounds corresponding to the
intersection of all possible sets (called the kernel) and the union of all possible sets (called the envelope)
which are the possible candidate values for the variable. The consistency achieved on SetVariables is
therefore a kind of bound consistency.
constructors:
Choco method
makeSetVar(String name, int lowB, int uppB, String... options)
makeSetVarArray(String name, int dim, int lowB, int uppB, String... options)
return type
SetVariable
SetVariable[]
options:
• no option : equivalent to option Options.V ENUM
• Options.V ENUM : to force Solver to create SetVariable with enumerated domain for the caridinality variable.
• Options.V BOUND : to force Solver to create SetVariable with bounded cardinality.
• Options.V NO DECISION : to force variable to be removed from the pool of decisional variables.
• Options.V OBJECTIVE : to define the variable to be the one to optimize.
The variable representing the cardinality can be accessed and constrained using method getCard()
that returns an IntegerVariable object.
Example:
SetVariable svar1 = makeSetVar("svar1", -10, 10);
SetVariable svar2 = makeSetVar("svar2", 0, 10000, Options.V_BOUND, Options.V_NO_DECISION
);
Set variables are illustrated on the ternary Steiner problem.
6.4
Task variables
TaskVariable is an object-variable composed of three IntegerVariable: a starting time integer variable start, an ending time integer variable end and a duration integer variable duration. To create a
TaskVariable, one can creates the start, end and duration before, or indicates the earliest starting time
(int), the latest completion time (int) and the duration(int or IntegerVariable).
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CHAPTER 6. VARIABLES (MODEL)
constructors:
Choco method
makeTaskVar(String name, IntegerVariable start, IntegerVariable end,
IntegerVariable duration, String... options)
makeTaskVar(String name, IntegerVariable start, IntegerVariable duration,
String... options)
makeTaskVar(String name, int binf, int bsup, IntegerVariable duration,
String... options)
makeTaskVar(String name, int binf, int bsup, int duration, String... options
)
makeTaskVar(String name, int bsup, IntegerVariable duration, String...
options)
makeTaskVar(String name, int bsup, int duration, String... options)
makeTaskVarArray(String prefix, IntegerVariable[] starts, IntegerVariable[]
ends, IntegerVariable[] durations, String... options)
makeTaskVarArray(String name, int binf, int bsup, IntegerVariable[]
durations, String... options)
makeTaskVarArray(String name, int binf, int bsup, int[] durations, String...
options)
makeTaskVarArray(String name, int binf, int bsup, IntegerVariable[][]
durations, String... options)
makeTaskVarArray(String name, int binf, int bsup, int[][] durations, String
... options)
return type
TaskVariable
TaskVariable
TaskVariable
TaskVariable
TaskVariable
TaskVariable
TaskVariable[]
TaskVariable[]
TaskVariable[][]
TaskVariable[][]
TaskVariable[][]
options:
Options are for the three IntegerVariable. See IntegerVariable for more details about options.
Example:
TaskVariable tvar1 = makeTaskVar("tvar1", 0, 123, 18, Options.V_ENUM);
IntegerVariable start = makeIntVar("start", 0, 30);
IntegerVariable end = makeIntVar("end", 10, 60);
IntegerVariable duration = makeIntVar("duration", 7, 13);
TaskVariable tvar2 = makeTaskVar("tvar2", start, end, duration);
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Chapter 7
Operators (Model)
This section lists and details the operators that can be used within a Choco Model to combine variables
in expressions.
7.1
abs (operator)
Returns an expression variable that represents the absolute value of the argument (|n|).
• API : abs(IntegerExpressionVariable n)
• return type : IntegerExpressionVariable
• options : n/a
• favorite domain : unknown
Example:
Model m = new CPModel();
IntegerVariable x = makeIntVar("x", 1, 5, Options.V_ENUM);
IntegerVariable y = makeIntVar("y", -5, 5, Options.V_ENUM);
m.addConstraint(eq(abs(x), y));
Solver s = new CPSolver();
s.read(m);
s.solve();
7.2
cos (operator)
Returns an expression variable corresponding to the cosinus value of the argument (cos(x)).
• API : cos(RealExpressionVariable exp)
• return type : RealExpressionVariable
• options : n/a
• favorite domain : real
Example:
Model m = new CPModel();
RealVariable x = makeRealVar("x", -Math.PI/2, Math.PI);
m.addConstraint(eq(cos(x), 2/3));
Solver s = new CPSolver();
s.read(m);
s.solve();
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CHAPTER 7. OPERATORS (MODEL)
7.3
div (operator)
Returns an expression variable that represents the integer quotient of the division of the first argument
variable by the second one (n1 /n2 ).
• API :
– div(IntegerExpressionVariable n1, IntegerExpressionVariable n2)
– div(IntegerExpressionVariable n1, int n2)
– div(int n1, IntegerExpressionVariable n2)
• return type : IntegerExpressionVariable
• options : n/a
• favorite domain : n/a
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable x = makeIntVar("x", 1, 10);
IntegerVariable w = makeIntVar("w", 22, 44);
IntegerVariable z = makeIntVar("z", 12, 21);
m.addConstraint(eq(z, div(w, x)));
s.read(m);
s.solve();
7.4
ifThenElse (operator)
ifThenElse(c, v1 , v2 ) states that if the constraint c is satisfied, it returns the second parameter v1 ,
otherwise it returns the third one v2 .
• API : ifThenElse(Constraint c, IntegerExpressionVariable v1, IntegerExpressionVariable v2)
• return type : IntegerExpressionVariable
• options : n/a
• favorite domain : unknown
Example:
Model m = new CPModel();
IntegerVariable x = makeIntVar("x", 1, 5);
IntegerVariable y = makeIntVar("y", 0, 10);
m.addConstraint(eq(y, ifThenElse(gt(x,2), mult(x,x), x)));
Solver s = new CPSolver();
s.read(m);
s.solveAll();
7.5
max (operator)
Returns an expression variable equals to the greater value of the argument (max(x1 , x2 , ..., xn )).
• API :
– max(IntegerExpressionVariable x1, IntegerExpressionVariable x2)
– max(int x1, IntegerExpressionVariable x2)
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7.6. MIN (OPERATOR)
– max(IntegerExpressionVariable x1, int x2)
– max(IntegerExpressionVariable[] x)
• return type: IntegerExpressionVariable
• options : n/a
• favorite domain : to complete
Example:
Model m = new CPModel();
m.setDefaultExpressionDecomposition(true);
IntegerVariable[] v = makeIntVarArray("v", 3, -3, 3);
IntegerVariable maxv = makeIntVar("max", -3, 3);
Constraint c = eq(maxv, max(v));
m.addConstraint(c);
Solver s = new CPSolver();
s.read(m);
s.solveAll();
7.6
min (operator)
Returns an expression variable equals to the smaller value of the argument (min(x1 , x2 , ..., xn )).
• API :
– min(IntegerExpressionVariable x1, IntegerExpressionVariable x2)
– min(int x1, IntegerExpressionVariable x2)
– min(IntegerExpressionVariable x1, int x2)
– min(IntegerExpressionVariable[] x)
• return type: IntegerExpressionVariable
• options : n/a
• favorite domain : to complete
Example:
Model m = new CPModel();
m.setDefaultExpressionDecomposition(true);
IntegerVariable[] v = makeIntVarArray("v", 3, -3, 3);
IntegerVariable minv = makeIntVar("min", -3, 3);
Constraint c = eq(minv, min(v));
m.addConstraint(c);
Solver s = new CPSolver();
s.read(m);
s.solveAll();
7.7
minus (operator)
Returns an expression variable that corresponding to the difference between the two arguments (x − y).
• API :
– minus(IntegerExpressionVariable x, IntegerExpressionVariable y)
– minus(IntegerExpressionVariable x, int y)
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CHAPTER 7. OPERATORS (MODEL)
– minus(int x, IntegerExpressionVariable y)
– minus(RealExpressionVariable x, RealExpressionVariable y)
– minus(RealExpressionVariable x, double y)
– minus(double x, RealExpressionVariable y)
• return type :
– IntegerExpressionVariable, if parameters are IntegerExpressionVariable
– RealExpressionVariable, if parameters are RealExpressionVariable
• options : n/a
• favorite domain : to complete
Example
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable a = makeIntVar("a", 0, 4);
m.addConstraint(eq(minus(a, 1), 2));
s.read(m);
s.solve();
7.8
mod (operator)
Returns an expression variable that represents the integer remainder of the division of the first argument
variable by the second one (x1 %x2 ).
• API:
– mod(IntegerExpressionVariable x1, IntegerExpressionVariable x2)
– mod(int x1, IntegerExpressionVariable x2)
– mod(IntegerExpressionVariable x1, int x2)
• return type : IntegerExpressionVariable
• options : n/a
• favorite domain : n/a
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable x = makeIntVar("x", 1, 10);
IntegerVariable w = makeIntVar("w", 22, 44);
m.addConstraint(eq(1, mod(w, x)));
s.read(m);
s.solve();
7.9
mult (operator)
Returns an expression variable that corresponding to the product of variables in argument (x ∗ y).
• API :
– mult(IntegerExpressionVariable x, IntegerExpressionVariable y)
– mult(IntegerExpressionVariable x, int y)
CHOCO solver documentation
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7.10. NEG (OPERATOR)
– mult(int x, IntegerExpressionVariable y)
– mult(RealExpressionVariable x, RealExpressionVariable y)
– mult(RealExpressionVariable x, double y)
– mult(double x, RealExpressionVariable y)
• return type :
– IntegerExpressionVariable, if parameters are IntegerExpressionVariable
– RealExpressionVariable, if parameters are RealExpressionVariable
• options : n/a
• favorite domain : to complete
Example
CPModel m = new CPModel();
IntegerVariable x = makeIntVar("x", -10, 10);
IntegerVariable z = makeIntVar("z", -10, 10);
IntegerVariable w = makeIntVar("w", -10, 10);
m.addVariables(x, z, w);
CPSolver s = new CPSolver();
// x >= z * w
Constraint exp = geq(x, mult(z, w));
m.setDefaultExpressionDecomposition(true);
m.addConstraint(exp);
s.read(m);
s.solveAll();
7.10
neg (operator)
Returns an expression variable that is the opposite of the expression integer variable in argument (−x).
• API : neg(IntegerExpressionVariable x)
• return type : IntegerExpressionVariable
• options : n/a
• favorite domain : n/a
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable x = makeIntVar("x", -10, 10);
IntegerVariable w = makeIntVar("w", -10, 10);
// -x = w - 20
m.addConstraint(eq(neg(x), minus(w, 20)));
s.read(m);
s.solve();
7.11
plus (operator)
Returns an expression variable that corresponding to the sum of the two arguments (x + y).
• API :
– plus(IntegerExpressionVariable x, IntegerExpressionVariable y)
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CHAPTER 7. OPERATORS (MODEL)
– plus(IntegerExpressionVariable x, int y)
– plus(int x, IntegerExpressionVariable y)
– plus(RealExpressionVariable x, RealExpressionVariable y)
– plus(RealExpressionVariable x, double y)
– plus(double x, RealExpressionVariable y)
• return type :
– IntegerExpressionVariable, if parameters are IntegerExpressionVariable
– RealExpressionVariable, if parameters are RealExpressionVariable
• options : n/a
• favorite domain : to complete
Example
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable a = makeIntVar("a", 0, 4);
// a + 1 = 2
m.addConstraint(eq(plus(a, 1), 2));
s.read(m);
s.solve();
7.12
power (operator)
Returns an expression variable that represents the first argument raised to the power of the second
argument (xy ).
• API :
– power(IntegerExpressionVariable x, IntegerExpressionVariable y)
– power(int x, IntegerExpressionVariable y)
– power(IntegerExpressionVariable x, int y)
– power(RealExpressionVariable x, int y)
• return type:
– IntegerExpressionVariable, if parameters are IntegerExpressionVariable
– RealExpressionVariable, if parameters are RealExpressionVariable
• option : n/a
• favorite domain : to complete
Example :
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable x = makeIntVar("x", 0, 10);
IntegerVariable y = makeIntVar("y", 2, 4);
IntegerVariable z = makeIntVar("z", 28, 80);
m.addConstraint(eq(z, power(x, y)));
s.read(m);
s.solve();
CHOCO solver documentation
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7.13. SCALAR (OPERATOR)
7.13
scalar (operator)
Return an integer expression that corresponds to the scalar product of coefficients array and variables
array (c1 ∗ x1 + c2 ∗ x2 + ... + cn ∗ xn ).
• API :
– scalar(int[] c, IntegerVariable[] x)
– scalar(IntegerVariable[] x, int[] c)
• return type : IntegerExpressionVariable
• options : n/a
• favorite domain : to complete
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable[] vars = makeIntVarArray("C", 9, 1, 10);
int[] coefficients = new int[]{1, 2, 3, 4, 5, 6, 7, 8, 9};
m.addConstraint(eq(165, scalar(coefficients, vars)));
s.read(m);
s.solve();
System.out.print("165 = (" + coefficients[0] + "*" + s.getVar(vars[0]).getVal()+")");
for (int i = 1; i < vars.length; i++) {
System.out.print(" + (" + coefficients[i] + "*" + s.getVar(vars[i]).getVal()+")");
}
System.out.println();
7.14
sin (operator)
Returns a real variable that corresponding to the sinus value of the argument (sin(x)).
• API : sin(RealExpressionVariable exp)
• return type : RealExpressionVariable
• options : n/a
• favorite domain : real
Example:
Model m = new CPModel();
RealVariable x = makeRealVar("x", 0, Math.PI);
m.addConstraint(eq(sin(x), 1));
Solver s = new CPSolver();
s.read(m);
s.solve();
7.15
sum (operator)
Return an integer expression that corresponds to the sum of the variables given in argument (x1 + x2 +
... + xn ).
• API: sum(IntegerVariable... lv)
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CHAPTER 7. OPERATORS (MODEL)
• return type : IntegerExpressionVariable
• options : n/a
• favorite domain : to complete
Example :
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable[] vars = makeIntVarArray("C", 10, 1, 10);
m.addConstraint(eq(99, sum(vars)));
s.read(m);
s.solve();
if(s.isFeasible()){
System.out.print("99 = " + s.getVar(vars[0]).getVal());
for (int i = 1; i < vars.length; i++) {
System.out.print(" + "+s.getVar(vars[i]).getVal());
}
System.out.println();
}
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Chapter 8
Constraints (Model)
This section lists and details the constraints currently available in Choco.
8.1
abs (constraint)
abs(x, y) states that x is the absolute value of y:
x = |y|
• API : abs(IntegerVariable x, IntegerVariable y)
• return type : Constraint
• options : n/a
• favorite domain : enumerated
Example:
Model m = new CPModel();
IntegerVariable x = makeIntVar("x", 1, 5, Options.V_ENUM);
IntegerVariable y = makeIntVar("y", -5, 5, Options.V_ENUM);
m.addConstraint(abs(x, y));
Solver s = new CPSolver();
s.read(m);
s.solve();
8.2
allDifferent (constraint)
allDifferent(hx1 , .., xn i) states that the arguments have pairwise distinct values:
xi 6= xj ,
∀ i 6= j
This constraint is the basis of any matching problems. Notice that the filtering algorithm (AC [Régin, 1994]
or BC [López-Ortiz et al., 2003]) depends on the nature (enumerated or bounded) of variables x.
• API :
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CHAPTER 8. CONSTRAINTS (MODEL)
– allDifferent(IntegerVariable... x)
– allDifferent(String options, IntegerVariable... x)
• return type : Constraint
• options :
– no option: if the domains of x are enumerated, the constraint refers to the alldifferent of
[Régin, 1994]; if they are bounded, a dedicated algorithm [López-Ortiz et al., 2003] for bound
propagation is used instead.
– Options.C ALLDIFFERENT AC for [Régin, 1994] implementation of arc consistency
– Options.C ALLDIFFERENT BC for [López-Ortiz et al., 2003] implementation of bound consistency
– Options.C ALLDIFFERENT CLIQUE for propagating the clique of differences
• favorite domain : enumerated for arc consistency, bounded for bound consistency.
• references :
– [Régin, 1994]: A filtering algorithm for constraints of difference in CSPs
– [López-Ortiz et al., 2003]: A fast and simple algorithm for bounds consistency of the alldifferent constraint
– global constraint catalog: alldifferent
Example:
int n = 8;
CPModel m = new CPModel();
IntegerVariable[] queens = new IntegerVariable[n];
IntegerVariable[] diag1 = new IntegerVariable[n];
IntegerVariable[] diag2 = new IntegerVariable[n];
for (int i = 0; i < n; i++) {
queens[i] = makeIntVar("Q" + i, 1, n);
diag1[i] = makeIntVar("D1" + i, 1, 2 * n);
diag2[i] = makeIntVar("D2" + i, -n + 1, n);
}
m.addConstraint(allDifferent(queens));
for (int i = 0; i < n; i++) {
m.addConstraint(eq(diag1[i], plus(queens[i], i)));
m.addConstraint(eq(diag2[i], minus(queens[i], i)));
}
m.addConstraint(Options.C_ALLDIFFERENT_CLIQUE, allDifferent(diag1));
m.addConstraint(Options.C_ALLDIFFERENT_CLIQUE, allDifferent(diag2));
// diagonal constraints
CPSolver s = new CPSolver();
s.read(m);
long tps = System.currentTimeMillis();
s.solveAll();
System.out.println("tps nreines1 " + (System.currentTimeMillis() - tps) + " nbNode " + s
.
getNodeCount());
8.3
among (constraint)
among(z, hx1 , .., xn i, s) states that z is the number of xi belonging to set s:
z = |{i | xi ∈ s}|
CHOCO solver documentation
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8.4. AND (CONSTRAINT)
• API:
– among(IntegerVariable z, IntegerVariable[] x, int[] v)
– among(IntegerVariable z, IntegerVariable[] x, SetVariable s)
• return type: Constraint
• [Bessière et al., 2005a]: Among, common and disjoint Constraints
• [Bessière et al., 2006]: Among, common and disjoint Constraints
• global constraint catalog: among
Example:
among with a collection of values
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable nvar = makeIntVar("v1", 1, 2);
IntegerVariable[] vars = Choco.makeIntVarArray("var", 10, 0, 10);
int[] values = new int[]{2, 3, 5};
m.addConstraint(among(nvar, vars, values));
s.read(m);
s.solve();
among with a set variable
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable nvar = makeIntVar("v1", 1, 2);
IntegerVariable[] vars = Choco.makeIntVarArray("var", 10, 0, 10);
SetVariable values = Choco.makeSetVar("s", 2, 6);
m.addConstraint(among(nvar, vars, values));
s.read(m);
s.solve();
8.4
and (constraint)
and(hC1 , .., Cn i) states that constraints in arguments are all satisfied:
C1 ∧ C2 ∧ . . . ∧ Cn
and(hb1 , .., bn i) states that booleans in arguments are all true:
(b1 = 1) ∧ (b2 = 1) ∧ . . . ∧ (bn = 1)
• API :
– and(Constraint... c)
– and(IntegerVariable... b)
• return type : Constraint
• options : n/a
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CHAPTER 8. CONSTRAINTS (MODEL)
• favorite domain : n/a
• references :
global constraint catalog: and
Examples:
• example1:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable v1 = makeIntVar("v1", 0, 1);
IntegerVariable v2 = makeIntVar("v2", 0, 1);
m.addConstraint(and(eq(v1, 1), eq(v2, 1)));
s.read(m);
s.solve();
• example2
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable[] vars = makeBooleanVarArray("b", 10);
m.addConstraint(and(vars));
s.read(m);
s.solve();
8.5
atMostNValue (constraint)
atMostNValue(z, hx1 , .., xn i) states that the number of distinct values occurring in collection x is
at most z:
z ≥ |hx1 , .., xn i|
• API : atMostNValue(IntegerVariable z, IntegerVariable[] x)
• return type : Constraint
• options : n/a
• favorite domain : n/a
• references :
– [Bessière et al., 2005b] Filtering algorithms for the NValue constraint
– global constraint catalog: atmost nvalue
Example:
Model m = new CPModel();
CPSolver s = new CPSolver();
IntegerVariable v1 = makeIntVar("v1", 1, 1);
IntegerVariable v2 = makeIntVar("v2", 2, 2);
IntegerVariable v3 = makeIntVar("v3", 3, 3);
IntegerVariable v4 = makeIntVar("v4", 3, 4);
IntegerVariable n = makeIntVar("n", 3, 3);
Constraint c = atMostNValue(n, new IntegerVariable[]{v1, v2, v3, v4});
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8.6. BOOLCHANNELING (CONSTRAINT)
m.addConstraint(c);
s.read(m);
s.solve();
8.6
boolChanneling (constraint)
boolChanneling(b, x, v) states that boolean b is true if and only if x has value v:
(b = 1)
⇐⇒
(x = v)
b is an indicator variable acting as an observer of value v. See also domainChanneling for observing
all the values of x.
• API : boolChanneling(IntegerVariable b, IntegerVariable x, int v)
• return type : Constraint
• options : n/a
• favorite domain : enumerated for x
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable bool = makeIntVar("bool", 0, 1);
IntegerVariable x = makeIntVar("x", 0, 5);
m.addConstraint(boolChanneling(bool, x, 4));
s.read(m);
s.solveAll();
8.7
clause (constraint)
−
+
−
−
+
clause(hb+
1 , .., bn i, hb1 , .., bm i) states that at least one boolean bi is true or one boolean bj is false.
n
_
(b+
i = 1) ∨
i=1
m
_
(b−
j = 0)
j=1
• API :
– clause(IntegerVariable[] bpos, IntegerVariable[] bneg)
– clause(String options, IntegerVariable[] bpos, IntegerVariable[] bneg)
• return type : Constraint
• options :
– no option default filtering
– Options.C CLAUSES ENTAIL ensures quick entailment tests
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CHAPTER 8. CONSTRAINTS (MODEL)
• favorite domain : n/a.
• references : global constraint catalog: clause or
Example:
CPModel mod = new CPModel();
CPSolver s = new CPSolver();
IntegerVariable[] vars = makeBooleanVarArray("b", 8);
IntegerVariable[] plits1 = new IntegerVariable[]{vars[0], vars[3], vars[4]};
IntegerVariable[] nlits1 = new IntegerVariable[]{vars[1], vars[2], vars[6]};
mod.addConstraint(clause(plits1, nlits1));
IntegerVariable[] plits2 = new IntegerVariable[]{vars[5], vars[3]};
IntegerVariable[] nlits2 = new IntegerVariable[]{vars[1], vars[4], vars[7]};
mod.addConstraint(clause(plits2, nlits2));
s.read(mod);
s.solveAll();
8.8
costRegular (constraint)
costRegular(z, hx1 , .., xn i, L(Π), hci,j i) states that sequence hx1 , .., xn i is a word belonging to the
regular language L(Π) and that z is its cost computed as the sum of the individual symbol weights
ci,xi :
n
X
ci,xi = z.
hx1 , .., xn i ∈ L(Π) ∧
i=1
Like regular, this constraint is useful for modelling sequencing rules in personnel scheduling and
rostering problems. Furthermore it allows to handle a linear counter (or cost) on the sequence. See
multiCostRegular to simultaneously handle several linear counters.
costRegular is the optimization variant of the regular constraint. Enforcing GAC is NP-Hard, then
the implemented algorithm [Demassey et al., 2006] achieves an intermediate AC-BC level of consistency.
Let Lx = L(Π) ∩ (D1 × . . . × Dn ) be the set of words of the language L(Π) that can be matched by
hx1 , .., xn i according to their current domains hD1 , .., Dn i, then:
• Arc Consistency is enforced over x regarding the language and the lower and upper bounds of z:
for each value v ∈ Di , there exists a word in Lx , with v as its i-th symbol and whose cost is between
the bounds of z.
• Bound Consistency is enforced over z regarding x and the language: the lower and upper bounds
of z are set as the minimum and maximum costs of any words in Lx .
In summary, costRegular(z, x, L(Π), c) dominates its decomposition regular(x, L(Π))∧equation(z, x, c).
Another decomposition proposed in [Beldiceanu et al., 2005] can easily be generated by introducing intermediary cost variables hz1 , .., zn i and state variables hq0 , .., qn i, then posting constraints in extension
on each tuple (qi−1 , xi , qi , zi ) with the Π transition table, and one linear sum z = z1 + · · · + zn . In terms
of consistency, the two approaches are incomparable (words with costs out of the bounds of z may not
be filtered by the decomposition, see examples in [Menana and Demassey, 2009]).
Several API exists for defining the regular language:
• With a deterministic finite automaton (DFA) Π weighted by a cost table c with two dimensions,
then c[i][j] is the cost of any transition in Π labeled by j when processing the i-th symbolPof a word:
it models the cost of assigning variable xi to value j. The constraint ensures that z = i c[i][xi ].
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8.8. COSTREGULAR (CONSTRAINT)
• With a DFA Π weighted by a cost table c with three dimensions, then c[i][j][s] is the cost of the
transition in Π outgoing from state s and labeled by j when processing the i-th symbol of a word:
it models the cost of assigning variable xi to value j if assignment
P sequence hx1 , .., xi−1 i reaches
state s when processed by Π. The constraint ensures that z = i c[i][xi ][si ] where hs0 , s1 , .., sn i is
the sequence of states encountered when recognizing hx1 , .., xn i in Π.
• With a weighted valued multi-graph G(Π) and a node s, then G(Π) must be a layered graph with
n + 1 layers and s be the unique node in layer 0. Such a graph defines a valued DFA, by setting
the arcs as the transitions, the arc values as the transition labels, the arc weights as the transition
costs, the nodes as the states, and the nodes in the last layer as the accepting states. Note that
this DFA recognizes only words of length n. The constraint ensures that z is the total weight of
the path in G(Π) produced when recognizing hx1 , .., xn i.
Automaton Π is encoded as an object of class FiniteAutomaton, whose API contains:
FiniteAutomaton();
FiniteAutomaton(String regularExpression);
int addState();
void setInitialState(int state);
void setFinal(int state);
void addTransition(int state1, int state2, int.. labels);
FiniteAutomaton union(FiniteAutomaton a);
FiniteAutomaton intersection(FiniteAutomaton a);
FiniteAutomaton complement();
void minimize();
int getNbStates();
void toDotty(String dotFileName);
• API :
– costRegular(IntegerVariable z, IntegerVariable[] x, FiniteAutomaton pi, int[][] c)
based on the ConstraintType.COSTREGULAR implementation
– costRegular(IntegerVariable z, IntegerVariable[] x, FiniteAutomaton pi, int[][][] c)
based on the ConstraintType.FASTCOSTREGULAR implementation
– costRegular(IntegerVariable z, IntegerVariable[] x, DirectedMultigraph<Node,Arc> g, Node
s)
based on the ConstraintType.FASTCOSTREGULAR implementation
• return type : Constraint
• options : n/a
• favorite domain : n/a
• references : [Demassey et al., 2006]: A Cost-Regular based hybrid column generation approach
Example:
Build the FiniteAutomaton manually by adding states and transitions:
// z counts the number of 2 followed by a 0 or a 1 in sequence x
IntegerVariable[] vars = makeIntVarArray("x", 10, 0, 2, Options.V_ENUM);
IntegerVariable z = makeIntVar("z", 3, 4, Options.V_BOUND);
FiniteAutomaton auto = new FiniteAutomaton();
// states
int start = auto.addState();
int end = auto.addState();
auto.setInitialState(start);
auto.setFinal(start);
auto.setFinal(end);
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CHAPTER 8. CONSTRAINTS (MODEL)
// transitions
auto.addTransition(start, start, 0, 1);
auto.addTransition(start, end, 2);
auto.addTransition(end, start, 2);
auto.addTransition(end, start, 0, 1);
// costs
int[][][] costs = new int[vars.length][3][auto.getNbStates()];
for (int i = 0; i < costs.length; i++) {
costs[i][0][end] = 1;
costs[i][1][end] = 1;
}
CPModel m = new CPModel();
m.addConstraint(costRegular(z, vars, auto, costs));
CPSolver s = new CPSolver();
s.read(m);
s.solveAll();
Build the FiniteAutomaton from a combination of several regular expressions:
IntegerVariable[] vars = makeIntVarArray("x", 28, 0, 2, Options.V_ENUM);
IntegerVariable z = makeIntVar("z", 0, 100, Options.V_BOUND);
// different rules are formulated as patterns that must NOT be matched by x
List<String> forbiddenRegExps = new ArrayList<String>();
// do not end with ’00’ if start with ’11’
forbiddenRegExps.add("11(0|1|2)*00");
// at most three consecutive 0
forbiddenRegExps.add("(0|1|2)*0000(0|1|2)*");
// no pattern ’112’ at position 5
forbiddenRegExps.add("(0|1|2){4}112(0|1|2)*");
// pattern ’12’ after a 0 or a sequence of 0
forbiddenRegExps.add("(0|1|2)*02(0|1|2)*");
forbiddenRegExps.add("(0|1|2)*01(0|1)(0|1|2)*");
// at most three 2 on consecutive even positions
forbiddenRegExps.add("(0|1|2)((0|1|2)(0|1|2))*2(0|1|2)2(0|1|2)2(0|1|2)*");
// a unique automaton is built as the complement language
// composed of all the forbidden patterns
FiniteAutomaton auto = new FiniteAutomaton();
for (String reg : forbiddenRegExps) {
FiniteAutomaton a = new FiniteAutomaton(reg);
auto = auto.union(a);
auto.minimize();
}
auto = auto.complement();
auto.minimize();
auto.toDotty("myForbiddenRules.dot");
System.out.println(auto.getNbStates() + " states");
// costs: count the number of 0 and of 1 at odd positions
int[][] costs = new int[vars.length][3];
for (int i = 1; i < costs.length; i += 2) {
costs[i][0] = 1;
costs[i][1] = 1;
}
CPModel m = new CPModel();
m.addConstraint(costRegular(z, vars, auto, costs));
CPSolver s = new CPSolver();
s.read(m);
s.minimize(s.getVar(z), true);
CHOCO solver documentation
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8.9. CUMULATIVE (CONSTRAINT)
System.out.println(s.solutionToString());
8.9
cumulative (constraint)
to be cleaned.
cumulative(start,duration,height,capacity) states that a set of tasks (defined by their starting times, finishing dates, durations and heights (or consumptions)) are executed on a cumulative
resource of limited capacity. That is, the total height of the tasks which are executed at any time
t does not exceed the capacity of the resource:
X
height[i] ≤ capacity, (∀ time t)
{i | start[i]≤t<start[i]+duration[i]}
The notion of task does not exist yet in Choco. The cumulative takes therefore as input, several
arrays of integer variables (of same size n) denoting the starting, duration, and height of each task.
When the array of finishing times is also specified, the constraint ensures that start[i] + duration[i]
= end[i] for all task i. As usual, a task is executed in the interval [start,end-1].
For further informations, see the section devoted to this constraint in the Choco Tutorial document.
• API :
– cumulative(IntegerVariable[] start, IntegerVariable[] end, IntegerVariable[] duration,
IntegerVariable[] height, IntegerVariable capa, String... options)
– cumulative(IntegerVariable[] start, IntegerVariable[] end, IntegerVariable[] duration,
int[] height, int capa, String... options)
– cumulative(IntegerVariable[] start, IntegerVariable[] duration, IntegerVariable[] height
, IntegerVariable capa, String... options)
• return type : Constraint
• options :
– no option
– SettingType.TASK INTERVAL.getOptionName() for fast task intervals
– SettingType.SLOW TASK INTERVAL.getOptionName() for slow task intervals
– SettingType.VILIM CEF ALGO.getOptionName() for Vilim theta lambda tree + lazy computation of the inner maximization of the edge finding rule of Van hentenrick and Mercier
– SettingType.VHM CEF ALGO N2K.getOptionName() for Simple n2 ∗ k algorithm (lazy for
R) (CalcEF – Van Hentenrick)
• favorite domain : n/a
• references :
– [Beldiceanu and Carlsson, 2002] A new multi-resource cumulatives constraint with negative
heights
– global constraint catalog: cumulative
Example:
CHOCO solver documentation
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CHAPTER 8. CONSTRAINTS (MODEL)
CPModel m = new CPModel();
// data
int n = 11 + 3; //number of tasks (include the three fake tasks)
int[] heights_data = new int[]{2, 1, 4, 2, 3, 1, 5, 6, 2, 1, 3, 1, 1, 2};
int[] durations_data = new int[]{1, 1, 1, 2, 1, 3, 1, 1, 3, 4, 2, 3, 1, 1};
// variables
IntegerVariable capa = constant(7);
IntegerVariable[] starts = makeIntVarArray("start", n, 0, 5, Options.V_BOUND);
IntegerVariable[] ends = makeIntVarArray("end", n, 0, 6, Options.V_BOUND);
IntegerVariable[] duration = new IntegerVariable[n];
IntegerVariable[] height = new IntegerVariable[n];
for (int i = 0; i < height.length; i++) {
duration[i] = constant(durations_data[i]);
height[i] = makeIntVar("height " + i, new int[]{0, heights_data[i]});
}
TaskVariable[] tasks = Choco.makeTaskVarArray("Task", starts, ends, duration);
IntegerVariable[] bool = makeIntVarArray("taskIn?", n, 0, 1);
IntegerVariable obj = makeIntVar("obj", 0, n, Options.V_BOUND, Options.V_OBJECTIVE);
//post the cumulative
m.addConstraint(cumulative("cumulative", tasks, height, constant(0), capa,
Options.C_CUMUL_TI));
//post the channeling to know if the task is scheduled or not
for (int i = 0; i < n; i++) {
m.addConstraint(boolChanneling(bool[i], height[i], heights_data[i]));
}
//state the objective function
m.addConstraint(eq(sum(bool), obj));
CPSolver s = new CPSolver();
s.read(m);
//set the fake tasks to establish the profile capacity of the ressource
try {
s.getVar(starts[0]).setVal(1);
s.getVar(ends[0]).setVal(2);
s.getVar(height[0]).setVal(2);
s.getVar(starts[1]).setVal(2);
s.getVar(ends[1]).setVal(3);
s.getVar(height[1]).setVal(1);
s.getVar(starts[2]).setVal(3);
s.getVar(ends[2]).setVal(4);
s.getVar(height[2]).setVal(4);
} catch (ContradictionException e) {
System.out.println("error, no contradiction expected at this stage");
}
// maximize the number of tasks placed in this profile
s.maximize(s.getVar(obj), false);
System.out.println("Objective : " + (s.getVar(obj).getVal() - 3));
for (int i = 3; i < starts.length; i++) {
if (s.getVar(height[i]).getVal() != 0)
System.out.println("[" + s.getVar(starts[i]).getVal() + " - "
+ (s.getVar(ends[i]).getVal() - 1) + "]:"
+ s.getVar(height[i]).getVal());
}
8.10
cumulativeMax (constraint)
Specific case of Cumulative, where the consumption is equal to 0.
• API :
CHOCO solver documentation
BSD licence 2012
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8.11. CUMULATIVEMIN (CONSTRAINT)
– cumulativeMax(String name, TaskVariable[] tasks, IntegerVariable[] heights, IntegerVariable
[] usages, IntegerVariable capacity, String... options)
– cumulativeMax(String name, TaskVariable[] tasks, IntegerVariable[] heights, IntegerVariable
capacity, String... options)
– cumulativeMax(TaskVariable[] tasks, int[] heights, int capacity, String... options)
• return type : Constraint
• options :
– no option
– SettingType.TASK INTERVAL.getOptionName() for fast task intervals
– SettingType.SLOW TASK INTERVAL.getOptionName() for slow task intervals
– SettingType.VILIM CEF ALGO.getOptionName() for Vilim theta lambda tree + lazy computation of the inner maximization of the edge finding rule of Van hentenrick and Mercier
– SettingType.VHM CEF ALGO N2K.getOptionName() for Simple n2 ∗ k algorithm (lazy for
R) (CalcEF – Van Hentenrick)
• favorite domain : n/a
8.11
cumulativeMin (constraint)
Specific case of Cumulative, where the capacity is infinite.
• API :
– cumulativeMin(String name, TaskVariable[] tasks, IntegerVariable[] heights, IntegerVariable
[] usages, IntegerVariable consumption, String... options)
– cumulativeMin(String name, TaskVariable[] tasks, IntegerVariable[] heights, IntegerVariable
consumption, String... options)
– cumulativeMin(TaskVariable[] tasks, int[] heights, int consumption, String... options)
• return type : Constraint
• options :
– no option
– SettingType.TASK INTERVAL.getOptionName() for fast task intervals
– SettingType.SLOW TASK INTERVAL.getOptionName() for slow task intervals
– SettingType.VILIM CEF ALGO.getOptionName() for Vilim theta lambda tree + lazy computation of the inner maximization of the edge finding rule of Van hentenrick and Mercier
– SettingType.VHM CEF ALGO N2K.getOptionName() for Simple n2 ∗ k algorithm (lazy for
R) (CalcEF – Van Hentenrick)
• favorite domain : n/a
CHOCO solver documentation
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CHAPTER 8. CONSTRAINTS (MODEL)
8.12
disjoint (constraint)
2
disjoint(hT11 , .., Tn1 i, hT12 , .., Tm
i) states that each pair of tasks (Ti1 , Tj2 ) is in disjunction i.e., the
processings of the two tasks do not overlap in time:
Ti1 .end ≤ Tj2 .start
∨
Tj2 .end ≤ Ti1 .start,
∀i = 1..n, j = 1..m
CHOCO only provides a decomposition with reified precedences because the coloured cumulative is
not available.
• API: disjoint(TaskVariable[] t1, TaskVariable[] t2)
• return type: Constraint[]
• favorite domain : n/a.
• references :
– global constraint catalog: disjoint tasks
Example:
Model m = new CPModel();
Solver s = new CPSolver();
TaskVariable[] tasks1 = Choco.makeTaskVarArray("Task1", 0, 10, new int[]{2, 5});
TaskVariable[] tasks2 = Choco.makeTaskVarArray("Task2", 0, 10, new int[]{3, 4});
m.addConstraints(disjoint(tasks1, tasks2));
s.read(m);
s.solve();
8.13
disjunctive (constraint)
to be cleaned.
disjunctive(start,duration) states that a set of tasks (defined by their starting times and
durations) are executed on a ddisjunctive resource, i.e. they do not overlap in time:
|{i | start[i] ≤ t < start[i] + duration[i]}| ≤ 1,
(∀ time t)
The notion of task does not exist yet in Choco. The disjunctive takes therefore as input arrays of
integer variables (of same size n) denoting the starting and duration of each task. When the array of
finishing times is also specified, the constraint ensures that start[i] + duration[i] = end[i] for all
task i. As usual, a task is executed in the interval [start,end-1].
• API :
– disjunctive(IntegerVariable[] start, int[] duration, String...options)
– disjunctive(IntegerVariable[] start, IntegerVariable[] duration, String... options)
– disjunctive(IntegerVariable[] start, IntegerVariable[] end, IntegerVariable[] duration
, String... options)
– disjunctive(IntegerVariable[] start, IntegerVariable[] end, IntegerVariable[] duration
, IntegerVariable uppBound, String... options)
CHOCO solver documentation
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8.14. DISTANCEEQ (CONSTRAINT)
• return type : Constraint
• options :
– no option
– SettingType.OVERLOAD CHECKING.getOptionName() overload checking rule ( O(n*log(n)),
Vilim), also known as task interval
– SettingType.NF NL.getOptionName() NotFirst/NotLast rule ( O(n*log(n)), Vilim) (recommended)
– SettingType.DETECTABLE PRECEDENCE.getOptionName() Detectable Precedence rule (
O(n*log(n)), Vilim)
– SettingType.EDGE FINDING D.getOptionName() disjunctive Edge Finding rule ( O(n*log(n)),
Vilim) (recommended)
– SettingType.DEFAULT FILTERING.getOptionName() use filtering algorithm proposed by
Vilim. nested loop, each rule is applied until it reach it fixpoint
– SettingType.VILIM FILTERING.getOptionName() use filtering algorithm proposed by Vilim.
nested loop, each rule is applied until it reach it fixpoint
– SettingType.SINGLE RULE FILTERING.getOptionName() use filtering algorithm proposed
by Vilim. nested loop, each rule is applied until it reach it fixpoint. A single filtering rule
(debug only).
• favorite domain : n/a
• references :
global constraint catalog: disjunctive
Example: //TODO: complete
8.14
distanceEQ (constraint)
distanceEQ(x1 , x2 , x3 , c) states that x3 plus an offset c (by default c = 0) is equal to the distance
between x1 and x2 :
x3 + c = |x1 − x2 |
• API :
– distanceEQ(IntegerVariable x1, IntegerVariable x2, int x3)
– distanceEQ(IntegerVariable x1, IntegerVariable x2, IntegerVariable x3)
– distanceEQ(IntegerVariable x1, IntegerVariable x2, IntegerVariable x3, int c)
• return type: Constraint
• options : n/a
• favorite domain : to complete
• references :
global constraint catalog: all min dist (variant)
Example:
CHOCO solver documentation
BSD licence 2012
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CHAPTER 8. CONSTRAINTS (MODEL)
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable v0 = makeIntVar("v0", 0, 5);
IntegerVariable v1 = makeIntVar("v1", 0, 5);
IntegerVariable v2 = makeIntVar("v2", 0, 5);
m.addConstraint(distanceEQ(v0, v1, v2, 0));
s.read(m);
s.solveAll();
8.15
distanceGT (constraint)
distanceGT(x1 , x2 , x3 , c) states that x3 plus an offset c (by default c = 0) is strictly greater than
the distance between x1 and x2 :
x3 + c > |x1 − x2 |
• API :
– distanceGT(IntegerVariable x1, IntegerVariable x2, int x3)
– distanceGT(IntegerVariable x1, IntegerVariable x2, IntegerVariable x3)
– distanceGT(IntegerVariable x1, IntegerVariable x2, IntegerVariable x3, int c)
• return type: Constraint
• options : n/a
• favorite domain : to complete
• references :
global constraint catalog: all min dist (variant)
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable v0 = makeIntVar("v0", 0, 5);
IntegerVariable v1 = makeIntVar("v1", 0, 5);
IntegerVariable v2 = makeIntVar("v2", 0, 5);
m.addConstraint(distanceGT(v0, v1, v2, 0));
s.read(m);
s.solveAll();
8.16
distanceLT (constraint)
distanceLT(x1 , x2 , x3 , c) states that x3 plus an offset c (by default c = 0) is strictly smaller than
the distance between x1 and x2 :
x3 + c < |x1 − x2 |
• API :
CHOCO solver documentation
BSD licence 2012
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2/8/2012
8.17. DISTANCENEQ (CONSTRAINT)
– distanceLT(IntegerVariable x1, IntegerVariable x2, int x3)
– distanceLT(IntegerVariable x1, IntegerVariable x2, IntegerVariable x3)
– distanceLT(IntegerVariable x1, IntegerVariable x2, IntegerVariable x3, int c)
• return type: Constraint
• options : n/a
• favorite domain : to complete
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable v0 = makeIntVar("v0", 0, 5);
IntegerVariable v1 = makeIntVar("v1", 0, 5);
IntegerVariable v2 = makeIntVar("v2", 0, 5);
m.addConstraint(distanceLT(v0, v1, v2, 0));
s.read(m);
s.solveAll();
8.17
distanceNEQ (constraint)
distanceNEQ(x1 , x2 , x3 , c) states that x3 plus an offset c (by default c = 0) is not equal to the
distance between x1 and x2 :
x3 + c 6= |x1 − x2 |
• API :
– distanceNEQ(IntegerVariable x1, IntegerVariable x2, int x3)
– distanceNEQ(IntegerVariable x1, IntegerVariable x2, IntegerVariable x3)
– distanceNEQ(IntegerVariable x1, IntegerVariable x2, IntegerVariable x3, int c)
• return type: Constraint
• options : n/a
• favorite domain : to complete
• references :
global constraint catalog: all min dist (variant)
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable v0 = makeIntVar("v0", 0, 5);
IntegerVariable v1 = makeIntVar("v1", 0, 5);
m.addConstraint(distanceNEQ(v0, v1, 0));
s.read(m);
s.solveAll();
CHOCO solver documentation
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2/8/2012
CHAPTER 8. CONSTRAINTS (MODEL)
8.18
domainChanneling (constraint)
domainChanneling(x, hb1 , .., bn i) states that boolean bj is true if and only if x has value j:
bj = 1
⇐⇒
x = j,
∀j = 1..n
It makes the link between a domain variable x and those 0-1 variables b that are associated with each
potential value of x: the 0-1 variable bj associated with the value j taken by x is equal to 1, while the
remaining 0-1 variables bi (i 6= j) are all equal to 0.
• API : domainChanneling(IntegerVariable x, IntegerVariable[] b)
• return type : Constraint
• options : n/a
• favorite domain : enumerated for x
• references :
global constraint catalog: domain constraint
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable x = makeIntVar("var", 0, 10);
IntegerVariable[] b = makeBooleanVarArray("valueIndicator", 10);
m.addConstraint(domainChanneling(x, b));
s.read(m);
s.solveAll();
8.19
element (constraint)
See nth.
8.20
endsAfter (constraint)
endsAfter(T, c) states that task T ends after time c:
T.end ≥ c
• API :endsAfter(TaskVariable t, int c)
• return type : Constraint
• options : n/a
• favorite domain : n/a.
Examples: to complete
CHOCO solver documentation
BSD licence 2012
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8.21. ENDSAFTERBEGIN (CONSTRAINT)
8.21
endsAfterBegin (constraint)
endsAfterBegin(T1 , T2 , c) states that task T1 ends after the start time of T2 minus c:
T1 .end ≥ T2 .start − c
• API :endsAfterBegin(TaskVariable t1, TaskVariable t2, int c)
• return type : Constraint
• options : n/a
• favorite domain : n/a.
Examples: to complete
8.22
endsAfterEnd (constraint)
endsAfterEnd(T1 , T2 , c) states that task T1 ends after the end time of T2 minus c:
T1 .end ≥ T2 .end − c
• API :endsAfterEnd(TaskVariable t1, TaskVariable t2, int c)
• return type : Constraint
• options : n/a
• favorite domain : n/a.
Examples: to complete
8.23
endsBefore (constraint)
endsBefore(T, c) states that task T ends before time c:
T.end ≤ c
• API :endsBefore(final TaskVariable t, final int c)
• return type : Constraint
• options : n/a
• favorite domain : n/a.
Examples: to complete
CHOCO solver documentation
BSD licence 2012
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2/8/2012
CHAPTER 8. CONSTRAINTS (MODEL)
8.24
endsBeforeBegin (constraint)
endsBeforeBegin(T1 , T2 , c) states that task T1 ends before the start time of T2 minus c:
T1 .end ≤ T2 .start − c
• API :endsBeforeBegin(TaskVariable t1, TaskVariable t2, int c)
• return type : Constraint
• options : n/a
• favorite domain : n/a.
Examples: to complete
8.25
endsBeforeEnd (constraint)
endsBeforeEnd(T1 , T2 , c) states that task T1 ends before the end time of T2 minus c:
T1 .end ≤ T2 .end − c
• API :endsBeforeEnd(TaskVariable t1, TaskVariable t2, int c)
• return type : Constraint
• options : n/a
• favorite domain : n/a.
Examples: to complete
8.26
endsBetween (constraint)
endsBetween(T, c1 , c2 ) states that task T ends between times c1 and c2 :
c1 ≤ T.end ≤ c2
• API :endsBetween(TaskVariable t, int min, int max)
• return type : Constraint
• options : n/a
• favorite domain : n/a.
Examples: to complete
CHOCO solver documentation
BSD licence 2012
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2/8/2012
8.27. EQ (CONSTRAINT)
8.27
eq (constraint)
eq(x, y) states that the two arguments are equal:
x=y
• API :
– eq(IntegerExpressionVariable x, IntegerExpressionVariable y)
– eq(IntegerExpressionVariable x, int y)
– eq(int x, IntegerExpressionVariable y)
– eq(SetVariable x, SetVariable y)
– eq(RealExpressionVariable x, RealExpressionVariable y)
– eq(RealExpressionVariable x, double y)
– eq(double x, RealExpressionVariable y)
– eq(IntegerVariable x, RealVariable y)
– eq(RealVariable x, IntegerVariable y)
• return type : Constraint
• options : n/a
• favorite domain : to complete.
• references :
global constraint catalog: eq (on domain variables) and eq set (on set variables).
Examples:
• example1:
Model m = new CPModel();
Solver s = new CPSolver();
int c = 1;
IntegerVariable v = makeIntVar("v", 0, 2);
m.addConstraint(eq(v, c));
s.read(m);
s.solve();
• example2
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable v1 = makeIntVar("v1", 0, 2);
IntegerVariable v2 = makeIntVar("v2", 0, 2);
IntegerExpressionVariable w1 = plus(v1, 1);
IntegerExpressionVariable w2 = minus(v2, 1);
m.addConstraint(eq(w1, w2));
s.read(m);
s.solve();
CHOCO solver documentation
BSD licence 2012
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2/8/2012
CHAPTER 8. CONSTRAINTS (MODEL)
8.28
eqCard (constraint)
eqCard(s, z) states that the cardinality of set s is equal to z:
|s| = z
• API :
– eqCard(SetVariable s, IntegerVariable z)
– eqCard(SetVariable s, int z)
• return type : Constraint
• options : n/a
• favorite domain : to complete
Example:
Model m = new CPModel();
Solver s = new CPSolver();
SetVariable set = makeSetVar("s", 1, 5);
IntegerVariable card = makeIntVar("card", 2, 3);
m.addConstraint(member(set, 3));
m.addConstraint(eqCard(set, card));
s.read(m);
s.solve();
8.29
equation (constraint)
equation(z, hx1 , .., xn i, hc1 , .., cn i) states that z is the weighted sum of x by c:
c1 x1 + c2 x2 + ... + cn xn = z
Restrictions: {∀ci ∈ c, ci ≥ 0} ∧ z ≥ 0
See also knapsackProblem.
• API :
– equation(int z, IntegerVariable[] x, int[] c)
– equation(String option, int z, IntegerVariable[] x, int[] c)
– equation(IntegerVariable z, IntegerVariable[] x, int[] c)
– equation(String option, IntegerVariable z, IntegerVariable[] x, int[] c)
• return type : Constraint
• options :
– no option or "cp:ac": to enforce GAC using regular
– "cp:bc": to enforce bound consistency using eq(z,scalar(x, c))
• favorite domain : to complete
CHOCO solver documentation
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2/8/2012
8.30. EXACTLY (CONSTRAINT)
• global constraint catalog : scalar product
Example:
CPModel m = new CPModel();
CPSolver s = new CPSolver();
int n = 10;
IntegerVariable[] bvars = makeIntVarArray("b", n, 0, 10, Options.V_ENUM);
int[] coefs = new int[n];
int charge = 10;
Random rand = new Random();
for (int i = 0; i < coefs.length; i++) {
coefs[i] = rand.nextInt(10);
}
Constraint knapsack = equation(charge, bvars, coefs);
m.addConstraint(knapsack);
s.read(m);
s.solveAll();
8.30
exactly (constraint)
Deprecated, see occurrence.
8.31
FALSE (constraint)
F ALSE always returns false.
8.32
feasPairAC (constraint)
feasPairAC(x, y, f easT uples) states an extensional binary constraint on (x, y) defined by the table
f easT uples of compatible pairs of values, and then enforces arc consistency. Two APIs are available
to define the compatible pairs:
• if f easT uples is encoded as a list of pairs List<int[2]>, then:
∃ tuple i | (x, y) = f easT uples[i]
• if f easT uples is encoded as a boolean matrix boolean[][], let x and y be the initial minimum
values of x and y, then:
∃(u, v) | (x, y) = (u + x, v + y) ∧ f easT uples[u][v]
The two APIs are duplicated to allow definition of options.
• API :
– feasPairAC(IntegerVariable x, IntegerVariable y, List<int[]> feasTuples)
– feasPairAC(String options, IntegerVariable x, IntegerVariable y, List<int[]> feasTuples
)
– feasPairAC(IntegerVariable x, IntegerVariable y, boolean[][] feasTuples)
– feasPairAC(String options, IntegerVariable x, IntegerVariable y, boolean[][] feasTuples
)
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CHAPTER 8. CONSTRAINTS (MODEL)
• return type : Constraint
• options :
– no option: use AC3 (default arc consistency)
– Options.C EXT AC3: to get AC3 algorithm (searching from scratch for supports on all values)
– Options.C EXT AC2001: to get AC2001 algorithm (maintaining the current support of each
value)
– Options.C EXT AC32: to get AC3rm algorithm (maintaining the current support of each value
in a non backtrackable way)
– Options.C EXT AC322: to get AC3 with the used of BitSet to know if a support still exists
• favorite domain : to complete
• references :
global constraint catalog: in relation
Example:
Model m = new CPModel();
Solver s = new CPSolver();
ArrayList<int[]> couples2 = new ArrayList<int[]>();
couples2.add(new int[]{1, 2});
couples2.add(new int[]{1, 3});
couples2.add(new int[]{2, 1});
couples2.add(new int[]{3, 1});
couples2.add(new int[]{4, 1});
IntegerVariable v1 = makeIntVar("v1", 1, 4);
IntegerVariable v2 = makeIntVar("v2", 1, 4);
m.addConstraint(feasPairAC(Options.C_EXT_AC32, v1, v2, couples2));
s.read(m);
s.solveAll();
8.33
feasTupleAC (constraint)
feasTupleAC(hx1 , .., xn i, f easT uples) states an extensional constraint on hx1 , .., xn i defined by the
table f easT uples of compatible tuples of values, and then enforces arc consistency:
∃ tuple i |
hx1 , .., xn i = f easT uples[i]
The API is duplicated to define options.
• API :
– feasTupleAC(List<int[]> feasTuples, IntegerVariable... x)
– feasTupleAC(String options, List<int[]> feasTuples, IntegerVariable... x)
• return type: Constraint
• options :
– no option: use AC32 (default arc consistency)
– Options.C EXT AC32: to get AC3rm algorithm (maintaining the current support of each value
in a non backtrackable way)
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8.34. FEASTUPLEFC (CONSTRAINT)
– Options.C EXT AC2001: to get AC2001 algorithm (maintaining the current support of each
value)
– Options.C EXT AC2008: to get AC2008 algorithm (maintained by STR)
• favorite domain : to complete
• references :
global constraint catalog: in relation
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable v1 = makeIntVar("v1", 0, 2);
IntegerVariable v2 = makeIntVar("v2", 0, 4);
ArrayList<int[]> feasTuple = new ArrayList<int[]>();
feasTuple.add(new int[]{1, 1}); // x*y = 1
feasTuple.add(new int[]{2, 4}); // x*y = 1
m.addConstraint(feasTupleAC(Options.C_EXT_AC2001, feasTuple, v1, v2));
s.read(m);
s.solve();
8.34
feasTupleFC (constraint)
feasTupleFC(hx1 , .., xn i, f easT uples) states an extensional constraint on hx1 , .., xn i defined by the
table f easT uples of compatible tuples of values, and then performs Forward Checking:
∃ tuple i |
hx1 , .., xn i = f easT uples[i]
• API : feasTupleFC(List<int[]> tuples, IntegerVariable... x)
• return type: Constraint
• options : n/a
• favorite domain: to complete
• references :
global constraint catalog: in relation
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable v1 = makeIntVar("v1", 0, 2);
IntegerVariable v2 = makeIntVar("v2", 0, 4);
ArrayList<int[]> feasTuple = new ArrayList<int[]>();
feasTuple.add(new int[]{1, 1}); // x*y = 1
feasTuple.add(new int[]{2, 4}); // x*y = 1
m.addConstraint(feasTupleFC(feasTuple, v1, v2));
s.read(m);
s.solve();
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CHAPTER 8. CONSTRAINTS (MODEL)
8.35
forbiddenInterval (constraint)
to be detailed
forbiddenInterval(ht1 , .., t2 i) applies additionnal search tree reduction based on time intervals
in which no operation can start or end in an optimal solution. The tasks must all belong to one
disjunctive resource and have fixed durations.
• API :
– forbiddenInterval(String name, TaskVariable[] tasks)
– forbiddenInterval(TaskVariable[] tasks)
• return type : Constraint
• options : n/a
• favorite domain : n/a
Example: to complete
8.36
geost (constraint)
to be cleaned
geost is a global constraint that generically handles a variety of geometrical placement problems.
It handles geometrical constraints (non-overlapping, distance, etc.) between polymorphic objects
(ex: polymorphism can be used for representing rotation) in any dimension. The parameters of
geost(dim, objects, shif tedBoxes, eCtrs) are respectively: the space dimension, the list of geometrical objects, the set of boxes that compose the shapes of the objects, the set of geometrical
constraints. The greedy mode should be used without external constraints to have safe results,
because it excludes external constraints from its exploration and look for instanciation of variables
involved in geost which respect the geost constraint. For further informations, see the section
devoted to this constraint in the Choco Tutorial document.
• API :
geost(int dim, Vector<GeostObject> objects, Vector<ShiftedBox> shiftedBoxes, Vector<ExternalConstraint
> eCtrs)
geost(int dim, Vector<GeostObject> objects, Vector<ShiftedBox> shiftedBoxes, Vector<ExternalConstraint
> eCtrs, Vector<int[]> ctrlVs)
• return type : Constraint
• options :n/a
• favorite domain : to complete
• references :
global constraint catalog: geost
The geost constraint requires the creation of different objects:
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8.36. GEOST (CONSTRAINT)
parameter
objects
shiftedBoxes
eCtrs
ctrlVs
Where a GeostObject
attribute
dim
objectId
shapeId
coordinates
startTime
durationTime
endTime
type
Vector<GeostObject>
Vector<ShiftedBox>
Vector<ExternalConstraint>
Vector<int[]>
is defined by:
type
int
int
IntegerVariable
IntegerVariable[dim]
IntegerVariable
IntegerVariable
IntegerVariable
description
geometrical objects
boxes that compose the object shapes
geometrical constraints
controlling vectors (for greedy mode)
description
dimension
object id
shape id
coordinates of the origin
starting time
duration
finishing time
Where a ShiftedBox is a dim-box defined by the shape it belongs to, its origin (the coordinates of the
lower left corner of the box) and its lengths in every dimensions:
attribute
type
description
sid
int
shape id
offset
int[dim]
coordinates of the offset (lower left corner)
size
int[dim]
lengths in every dimensions
Where an ExternalConstraint contains informations and functionality common to all external constraints and is defined by:
attribute
type
description
ectrID
int
constraint id
dimensions
int[]
list of dimensions that the external constraint
is active for
objectIdentifiers
int[]
list of object ids that this external constraint
affects.
Example:
Model m = new CPModel();
int dim = 3;
int lengths[] = {5, 3, 2};
int widths[] = {2, 2, 1};
int heights[] = {1, 1, 1};
int nbOfObj = 3;
long seed = 0;
//Create the Objects
List<GeostObject> obj = new ArrayList<GeostObject>();
for (int i = 0; i < nbOfObj; i++) {
IntegerVariable shapeId = Choco.makeIntVar("sid", i, i);
IntegerVariable coords[] = new IntegerVariable[dim];
for (int j = 0; j < coords.length; j++) {
coords[j] = Choco.makeIntVar("x" + j, 0, 2);
}
IntegerVariable start = Choco.makeIntVar("start", 1, 1);
IntegerVariable duration = Choco.makeIntVar("duration", 1, 1);
IntegerVariable end = Choco.makeIntVar("end", 1, 1);
obj.add(new GeostObject(dim, i, shapeId, coords, start, duration, end));
}
//Create the ShiftedBoxes and add them to corresponding shapes
List<ShiftedBox> sb = new ArrayList<ShiftedBox>();
int[] t = {0, 0, 0};
for (int d = 0; d < nbOfObj; d++) {
int[] l = {lengths[d], heights[d], widths[d]};
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CHAPTER 8. CONSTRAINTS (MODEL)
sb.add(new ShiftedBox(d, t, l));
}
//Create the external constraints vector
List<IExternalConstraint> ectr = new ArrayList<IExternalConstraint>();
//create the list of dimensions for the external constraint
int[] ectrDim = new int[dim];
for (int d = 0; d < dim; d++)
ectrDim[d] = d;
//create the list of object ids for the external constraint
int[] objOfEctr = new int[nbOfObj];
for (int d = 0; d < nbOfObj; d++) {
objOfEctr[d] = obj.get(d).getObjectId();
}
//create and add one external constraint of type non overlapping
NonOverlappingModel n = new NonOverlappingModel(Constants.NON_OVERLAPPING, ectrDim,
objOfEctr);
ectr.add(n);
//create and post the geost constraint
Constraint geost = Choco.geost(dim, obj, sb, ectr);
m.addConstraint(geost);
Solver s = new CPSolver();
s.read(m);
s.setValIntSelector(new RandomIntValSelector(seed));
s.setVarIntSelector(new RandomIntVarSelector(s, seed));
s.solveAll();
8.37
geq (constraint)
geq(x, y) states that x is greater than or equal to y:
x≥y
• API :
– geq(IntegerExpressionVariable x, IntegerExpressionVariable y)
– geq(IntegerExpressionVariable x, int y)
– geq(int x, IntegerExpressionVariable y)
– geq(RealExpressionVariable x, RealExpressionVariable y)
– geq(RealExpressionVariable x, double y)
– geq(double x, RealExpressionVariable y)
• return type : Constraint
• options : n/a
• favorite domain : to complete.
• references :
global constraint catalog: geq
Examples:
• example1:
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8.38. GEQCARD (CONSTRAINT)
Model m = new CPModel();
Solver s = new CPSolver();
int c = 1;
IntegerVariable v = makeIntVar("v", 0, 2);
m.addConstraint(eq(v, c));
s.read(m);
s.solve();
• example2
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable v1 = makeIntVar("v1", 0, 2);
IntegerVariable v2 = makeIntVar("v2", 0, 2);
IntegerExpressionVariable w1 = plus(v1, 1);
IntegerExpressionVariable w2 = minus(v2, 1);
m.addConstraint(eq(w1, w2));
s.read(m);
s.solve();
8.38
geqCard (constraint)
geqCard(s, z) states that the cardinality of set s is greater than or equal to z:
|s| ≥ z
• API :
– geqCard(SetVariable s, IntegerVariable z)
– geqCard(SetVariable s, int z)
• return type : Constraint
• options : n/a
• favorite domain : to complete
Example:
Model m = new CPModel();
Solver s = new CPSolver();
SetVariable set = makeSetVar("s", 1, 5);
IntegerVariable i = makeIntVar("card", 2, 3);
m.addConstraint(member(set, 3));
m.addConstraint(geqCard(set, i));
s.read(m);
s.solve();
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CHAPTER 8. CONSTRAINTS (MODEL)
8.39
globalCardinality (constraint)
globalCardinality(hx1 , .., xn i, hl0 , .., lM −m i, hu0 , .., uM −m i, m) states lower bounds l and upper
bounds u on the occurrence numbers of the values in collection x according to offset m:
lj−m ≤ |{i = 1..n , xi = j}| ≤ uj−m ,
∀j = m..M
globalCardinality(hx1 , .., xn i, hz0 , .., zM −m i, o) states that z are the occurrence numbers of the
values in collection x according to offset m:
zj−m = |{i = 1..n , xi = j}|,
∀j = m..M
Note that the length of the bound tables l and u are equal to M + m − 1 where m, the offset, is the
minimum counted value and M is the maximum counted value.
Several APIs exist:
• constant bounds on cardinalities hl0 , .., lM −m i and hu0 , .., uM −m i: use the propagator of [Régin, 1996]
or of [Quimper et al., 2003] depending on the set options and the nature of the domain variables.
• variable cardinalities hz0 , .., zM −m i: use the propagator of [Quimper et al., 2003] that:
– enforces Bound Consistency over x regarding the lower and upper bounds of z,
– maintains the upper bound of zj by counting the variables that may be instantiated to j,
– maintains the lower bound of zj by counting the variables instantiated to j,
– enforces z0 + · · · + zM −m = n
The APIs are duplicated to define options.
• API :
– globalCardinality(IntegerVariable[] x, int[] low, int[] up, int offset)
– globalCardinality(String options, IntegerVariable[] x, int[] low, int[] up, int offset
)
– globalCardinality(IntegerVariable[] x, int[] values, int[] low, int[] up)
– globalCardinality(IntegerVariable[] x, IntegerVariable[] card, int offset)
– globalCardinality(IntegerVariable[] x, int[] values, IntegerVariable[] card)
• return type : Constraint
• options:
– no option: if x have bounded domains or if the cardinalities are variable z, use the propagator
of [Quimper et al., 2003] for BC, otherwise use the propagator of [Régin, 1996];
– Options.C GCC AC : for [Régin, 1996] implementation of arc consistency
– Options.C GCC BC : for [Quimper et al., 2003] implementation of bound consistency
• favorite domain : enumerated for arc consistency, bounded for bound consistency.
• references :
– [Régin, 1996]: Generalized arc consistency for global cardinality constraint,
– [Quimper et al., 2003]: An efficient bounds consistency algorithm for the global cardinality
constraint
– global constraint catalog: global cardinality
Examples:
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8.39. GLOBALCARDINALITY (CONSTRAINT)
• example1:
int n = 5;
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable[] vars = new IntegerVariable[n];
for (int i = 0; i < n; i++) {
vars[i] = makeIntVar("var " + i, 1, n);
}
int[] LB2 = {0, 1, 1, 0, 3};
int[] UB2 = {0, 1, 1, 0, 3};
m.addConstraint(Options.C_GCC_BC, globalCardinality(vars, LB2, UB2, 1));
s.read(m);
s.solve();
• example2:
int n = 5;
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable[] vars = makeIntVarArray("vars", n, 1, n);
IntegerVariable[] cards = makeIntVarArray("cards", n, 0, 1);
m.addConstraint(Options.C_GCC_BC, globalCardinality(vars, cards, 1));
s.read(m);
s.solve();
• example3:
int[] values = {1,3};
int[] occmin = {1, 2};
int[] occmax = {3, 2};
Model m = new CPModel();
IntegerVariable[] vars = makeIntVarArray("var", 7, 1, 4);
m.addConstraint(globalCardinality(vars, values, occmin, occmax));
Solver s = new CPSolver();
s.read(m);
s.solveAll();
• example4:
int[] values = {1,3};
Model m = new CPModel();
IntegerVariable[] vars = makeIntVarArray("var", 7, 1, 4);
IntegerVariable[] cards = makeIntVarArray("card", 2, 1, 3);
m.addConstraint(globalCardinality(vars, values, cards));
m.addConstraint(member(cards[0], new int[]{1, 2, 3}));
m.addConstraint(eq(cards[1], 2));
Solver s = new CPSolver();
s.read(m);
s.solveAll();
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CHAPTER 8. CONSTRAINTS (MODEL)
8.40
gt (constraint)
gt(x, y) states that x is strictly greater than y:
x>y
• API :
– gt(IntegerExpressionVariable x, IntegerExpressionVariable y)
– gt(IntegerExpressionVariable x, int y)
– gt(int x, IntegerExpressionVariable y)
• return type : Constraint
• options : n/a
• favorite domain : to complete.
• references :
global constraint catalog: gt
Example:
Model m = new CPModel();
Solver s = new CPSolver();
int c = 1;
IntegerVariable v = makeIntVar("v", 0, 2);
m.addConstraint(gt(v, c));
s.read(m);
s.solve();
8.41
ifOnlyIf (constraint)
ifOnlyIf(c1 , c2 ) states that c1 holds if and only if c2 holds:
c1 ⇐⇒ c2
• API : ifOnlyIf(Constraint c1, Constraint c2)
• return type : Constraint
• options : n/a
• favorite domain : n/a
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable x = makeIntVar("x", 1, 3);
IntegerVariable y = makeIntVar("y", 1, 3);
IntegerVariable z = makeIntVar("z", 1, 3);
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8.42. IFTHENELSE (CONSTRAINT)
m.addVariables(Options.V_BOUND,x ,y, z);
m.addConstraint(ifOnlyIf(lt(x, y), lt(y, z)));
s.read(m);
s.solveAll();
8.42
ifThenElse (constraint)
ifThenElse(c1 , c2 , c3 ) states that if c1 holds then c2 holds, otherwise c3 holds:
(c1 ∧ c2 ) ∨ (¬c1 ∧ c3 )
• API :ifThenElse(Constraint c1, Constraint c2, Constraint c3)
• return type : Constraint
• options : n/a
• favorite domain : n/a
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable x = makeIntVar("x", 1, 3);
IntegerVariable y = makeIntVar("y", 1, 3);
IntegerVariable z = makeIntVar("z", 1, 3);
// use API ifThenElse(Constraint, Constraint, Constraint)
m.addConstraint(ifThenElse(lt((x), (y)), gt((y), (z)), FALSE));
// and ifThenElse(Constraint, IntegerExpressionVariable, IntegerExpressionVariable)
m.addConstraint(leq(z, ifThenElse(lt(x, y), constant(1), plus(x,y))));
s.read(m);
s.solveAll();
8.43
implies (constraint)
implies(c1 , c2 ) states that if c1 holds then c2 holds:
c1 =⇒ c2
• API : implies(Constraint c1, Constraint c2)
• return type : Constraint
• options : n/a
• favorite domain : n/a
Example:
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CHAPTER 8. CONSTRAINTS (MODEL)
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable x = makeIntVar("x", 1, 2);
IntegerVariable y = makeIntVar("y", 1, 2);
IntegerVariable z = makeIntVar("z", 1, 2);
m.addVariables(Options.V_BOUND,x ,y, z);
Constraint e1 = implies(leq(x, y), leq(x, z));
m.addConstraint(e1);
s.read(m);
s.solveAll();
8.44
increasingNValue (constraint)
increasingNValue(z, hx1 , .., xn i) states that hx1 , .., xn i is sorted in increasing order and that z is
the number of distinct values occurring in x.
z = |{x1 , . . . , xn }|
∧
xi ≤ xi+1 , ∀i = 1..n.
• API :
– increasingNValue(IntegerVariable z, IntegerVariable[] x)
– increasingNValue(String option, IntegerVariable z, IntegerVariable[] x)
• return type : Constraint
• options :
– no option filter on lower bound and on upper bound
– Options.C INCREASING NVALUE ATLEAST filter on lower bound only
– Options.C INCREASING NVALUE ATMOST filter on upper bound only
– Options.C INCREASING NVALUE BOTH –default value– filter on lower bound and on upper
bound
• favorite domain : to complete
• references :
global constraint catalog: increasing nvalue
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable nval = makeIntVar("nval", 1, 3);
IntegerVariable[] variables = makeIntVarArray("vars", 6, 1, 4);
m.addConstraint(increasing_nvalue(Options.C_INCREASING_NVALUE_BOTH, nval, variables));
s.read(m);
s.solveAll();
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8.45. INCREASINGSUM (CONSTRAINT)
8.45
increasingSum (constraint)
increasingSum(hx1 , .., xn i, s) states that hx1 , .., xn i is sorted in increasing order and that s is equal
to the sum of x.
X
∀i = [1, n − 1] xi ≤ xi+1 ∧
xi = s
i=[1,n]
• API : increasingSum(IntegerVariable[] x, IntegerVariable s)
• return type : Constraint
• options : no options
• favorite domain : bounded for x and s
• references : –
Example:
Model m = new CPModel();
IntegerVariable[] res = new IntegerVariable[3];
res[0] = makeIntVar("x0", -2, 3);
res[1] = makeIntVar("x1", -3, 3);
res[2] = makeIntVar("x2", -3, 0);
IntegerVariable sum = makeIntVar("s", -3, 3);
m.addConstraint(Choco.increasingSum(res, sum));
Solver s = new CPSolver();
s.read(m);
s.solve();
8.46
infeasPairAC (constraint)
infeasPairAC(x, y, inf easT uples) states an extensional binary constraint on (x, y) defined by the
table inf easT uples of forbidden pairs of values, and then enforces arc consistency. Two APIs are
available to define the forbidden pairs:
• if inf easT uples is encoded as a list of pairs List<int[2]>, then:
∀ tuple i | (x, y) 6= inf easT uples[i]
• if inf easT uples is encoded as a boolean matrix boolean[][], let x and y be the initial
minimum values of x and y, then:
∀(u, v) | (x, y) = (u + x, v + y) ∨ ¬inf easT uples[u][v]
The two APIs are duplicated to allow definition of options.
• API :
– infeasPairAC(IntegerVariable x, IntegerVariable y, List<int[]> infeasTuples)
– infeasPairAC(String options, IntegerVariable x, IntegerVariable y, List<int[]> infeasTuples
)
– infeasPairAC(IntegerVariable x, IntegerVariable y, boolean[][] infeasTuples)
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CHAPTER 8. CONSTRAINTS (MODEL)
– infeasPairAC(String options, IntegerVariable x, IntegerVariable y, boolean[][] infeasTuples
)
• return type : Constraint
• options :
– no option: use AC3 (default arc consistency)
– Options.C EXT AC3: to get AC3 algorithm (searching from scratch for supports on all values)
– Options.C EXT AC2001: to get AC2001 algorithm (maintaining the current support of each
value)
– Options.C EXT AC32: to get AC3rm algorithm (maintaining the current support of each value
in a non backtrackable way)
– Options.C EXT AC322: to get AC3 with the used of BitSet to know if a support still exists
• favorite domain : to complete
Example:
Model m = new CPModel();
Solver s = new CPSolver();
boolean[][] matrice2 = new boolean[][]{
{false, true, true, false},
{true, false, false, false},
{false, false, true, false},
{false, true, false, false}};
IntegerVariable v1 = makeIntVar("v1", 1, 4);
IntegerVariable v2 = makeIntVar("v2", 1, 4);
m.addConstraint(feasPairAC(Options.C_EXT_AC32,v1, v2, matrice2));
s.read(m);
s.solveAll();
8.47
infeasTupleAC (constraint)
infeasTupleAC(hx1 , .., xn i, f easT uples) states an extensional constraint on hx1 , .., xn i defined by
the table inf easT uples of compatible tuples of values, and then enforces arc consistency:
∀ tuple i |
hx1 , .., xn i =
6 inf easT uples[i]
The API is duplicated to define options.
• API :
– infeasTupleAC(List<int[]> infeasTuples, IntegerVariable... x)
– infeasTupleAC(String options, List<int[]> infeasTuples, IntegerVariable... x)
• return type: Constraint
• options :
– no option: use AC32 (default arc consistency)
– Options.C EXT AC32: to get AC3rm algorithm (maintaining the current support of each value
in a non backtrackable way)
– Options.C EXT AC2001: to get AC2001 algorithm (maintaining the current support of each
value)
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8.48. INFEASTUPLEFC (CONSTRAINT)
– Options.C EXT AC2008: to get AC2008 algorithm (maintained by STR)
• favorite domain : to complete
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable x = makeIntVar("x", 1, 5);
IntegerVariable y = makeIntVar("y", 1, 5);
IntegerVariable z = makeIntVar("z", 1, 5);
ArrayList<int[]> forbiddenTuples = new ArrayList<int[]>();
forbiddenTuples.add(new int[]{1, 1, 1});
forbiddenTuples.add(new int[]{2, 2, 2});
forbiddenTuples.add(new int[]{2, 5, 3});
m.addConstraint(infeasTupleAC(forbiddenTuples, x, y, z));
s.read(m);
s.solveAll();
8.48
infeasTupleFC (constraint)
infeasTupleFC(hx1 , .., xn i, f easT uples) states an extensional constraint on hx1 , .., xn i defined by
the table inf easT uples of compatible tuples of values, and then performs Forward Checking:
∀ tuple i |
hx1 , .., xn i =
6 inf easT uples[i]
• API : infeasTupleFC(List<int[]> infeasTuples, IntegerVariable... x)
• return type: Constraint
• options : n/a
• favorite domain: to complete
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable x = makeIntVar("x", 1, 5);
IntegerVariable y = makeIntVar("y", 1, 5);
IntegerVariable z = makeIntVar("z", 1, 5);
ArrayList<int[]> forbiddenTuples = new ArrayList<int[]>();
forbiddenTuples.add(new int[]{1, 1, 1});
forbiddenTuples.add(new int[]{2, 2, 2});
forbiddenTuples.add(new int[]{2, 5, 3});
m.addConstraint(infeasTupleFC(forbiddenTuples, x, y, z));
s.read(m);
s.solveAll();
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CHAPTER 8. CONSTRAINTS (MODEL)
8.49
intDiv (constraint)
intDiv(x, y, z) states that z is equal to the integer quotient of x by y:
z = bx/yc
• API: intDiv(IntegerVariable x, IntegerVariable y, IntegerVariable z)
• return type : Constraint
• option : n/a
• favorite domain: bound
Example:
Model m = new CPModel();
Solver s = new CPSolver();
long seed = 0;
IntegerVariable x = makeIntVar("x", 3, 5);
IntegerVariable y = makeIntVar("y", 1, 2);
IntegerVariable z = makeIntVar("z", 0, 5);
m.addConstraint(intDiv(x, y, z));
s.setVarIntSelector(new RandomIntVarSelector(s, seed));
s.setValIntSelector(new RandomIntValSelector(seed + 1));
s.read(m);
s.solve();
8.50
inverseChanneling (constraint)
inverseChanneling(hx1 , .., xn i, hy1 , .., ym i) states that xi has value j if and only if yj has value i:
xi = j
⇐⇒
yj = i,
∀i = 1..n, j = 1..m
• API : inverseChanneling(IntegerVariable[] x, IntegerVariable[] y)
• return type : Constraint
• options : no options
• favorite domain : enumerated for x and y
• references :
global constraint catalog: inverse
Example:
int n = 8;
Model m = new CPModel();
IntegerVariable[] queenInCol = new IntegerVariable[n];
IntegerVariable[] queenInRow = new IntegerVariable[n];
for (int i = 0; i < n; i++) {
queenInCol[i] = makeIntVar("QC" + i, 1, n);
queenInRow[i] = makeIntVar("QR" + i, 1, n);
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8.51. INVERSECHANNELINGWITHINRANGE (CONSTRAINT)
}
for (int i = 0; i < n; i++) {
for (int j = i + 1; j < n; j++) {
int k = j - i;
m.addConstraint(neq(queenInCol[i], queenInCol[j])); // row
m.addConstraint(neq(queenInCol[i], plus(queenInCol[j], k))); // diagonal 1
m.addConstraint(neq(queenInCol[i], minus(queenInCol[j], k))); // diagonal 2
m.addConstraint(neq(queenInRow[i], queenInRow[j])); // column
m.addConstraint(neq(queenInRow[i], plus(queenInRow[j], k))); // diagonal 2
m.addConstraint(neq(queenInRow[i], minus(queenInRow[j], k))); // diagonal 1
}
}
m.addConstraint(inverseChanneling(queenInCol, queenInRow));
Solver s = new CPSolver();
s.read(m);
s.solveAll();
8.51
inverseChannelingWithinRange (constraint)
inverseChannelingWithinRange(hx1 , .., xn i, hy1 , .., ym i) states that if xi is assigned to j and if j
is less than or equal to the number of items of the collection Y then yj is assigned to i. Conversely,
if yj is assigned to i and if i is less than or equal to the number of items of the collection X then xi
is assigned to j.
xi = j ∧ j < |y|
⇐⇒
yj = i ∧ i < |x|,
i ∈ [1, n], j ∈ [1, m]
• API : inverseChanneling(IntegerVariable[] x, IntegerVariable[] y)
• return type : Constraint
• options : no options
• favorite domain : enumerated for x and y
• references :
global constraint catalog: inverse within range
Example:
Model m = new CPModel();
IntegerVariable[] X = Choco.makeIntVarArray("X", 3, 0, 9, Options.V_ENUM);
IntegerVariable[] Y = Choco.makeIntVarArray("Y", 4, 0, 9, Options.V_ENUM);
m.addConstraint(Choco.eq(X[0], 9));
m.addConstraint(Choco.eq(Y[0], 9));
m.addConstraint(Choco.eq(Y[2], 9));
m.addConstraint(inverseChannelingWithinRange(X, Y));
Solver s = new CPSolver();
s.read(m);
s.solve();
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CHAPTER 8. CONSTRAINTS (MODEL)
8.52
inverseSet (constraint)
inverseSet(hx1 , .., xn i, hy1 , .., ym i) states that xi has (if an integer variable or contains, if a set
variable) value j if and only if yj contains value i:
If x is a collection of integer variables:
xi = j
⇐⇒
i ∈ sj ,
∀i = 0..n − 1, j = 0..m − 1
Notice that this version induces that y becomes a partition of the set of the indices of collection x.
If x is a collection of set variables:
j ∈ xi
⇐⇒
i ∈ sj ,
∀i = 0..n − 1, j = 0..m − 1
• API :
– inverseSet(IntegerVariable[] x, SetVariable[] y)
– inverseSet(SetVariable[] x, SetVariable[] y)
• return type : Constraint
• options : no options
• favorite domain : enumerated for x
• references :
global constraint catalog: inverse set
Example:
int i = 4;
int j = 2;
Model m = new CPModel();
IntegerVariable[] iv = makeIntVarArray("iv", i, 0, j);
SetVariable[] sv = makeSetVarArray("sv", j, 0, i);
m.addConstraint(inverseSet(iv, sv));
Solver s = new CPSolver();
s.read(m);
s.solveAll();
8.53
isIncluded (constraint)
isIncluded(s1 , s2 ) states that set s1 is included in set s2 :
s1 ⊆ s2
• API : isIncluded(SetVariable s1, SetVariable s2)
• return type : Constraint
• options :n/a
• favorite domain : to complete
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8.54. ISNOTINCLUDED (CONSTRAINT)
Example:
Model m = new CPModel();
Solver s = new CPSolver();
SetVariable v1 = makeSetVar("v1", 3, 4);
SetVariable v2 = makeSetVar("v2", 3, 8);
m.addConstraint(isIncluded(v1, v2));
s.read(m);
s.solveAll();
8.54
isNotIncluded (constraint)
isNotIncluded(s1 , s2 ) states that set s1 is not included in set s2 :
s1 6⊆ s2
• API : isNotIncluded(SetVariable s1, SetVariable s2)
• return type : Constraint
• options :n/a
• favorite domain : to complete
Example:
Model m = new CPModel();
Solver s = new CPSolver();
SetVariable v1 = makeSetVar("v1", 3, 4);
SetVariable v2 = makeSetVar("v2", 3, 8);
m.addConstraint(isNotIncluded(v1, v2));
s.read(m);
s.solveAll();
8.55
knapsackProblem (constraint)
knapsackProblem(z 1 , z 2 , hx1 , .., xn i, hc11 , .., c1n i, hc21 , .., c2n i) states that z 1 (respectively, z 2 ) is the sum
of the x weighted by the costs c1 (respectively, c2 ):
n
X
i=1
xi c1i = z 1
∧
n
X
xi c2i = z 2
i=1
The knaspack problem can be modeled using only this constraint and the objective maximize(z1):
xi is the number of items of type i and each item of type i has a value c1i and a weight c2i . Based
on costRegular, this propagator simulates the dynamic programming approach of [Trick, 2003]. It
dominates the filtering of the decomposition in two equation constraints.
• API : knapsackProblem(IntegerVariable z1, IntegerVariable z2, IntegerVariable[] x, int[] c1
, int[] c2)
• return type : Constraint
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CHAPTER 8. CONSTRAINTS (MODEL)
• options : n/a
• favorite domain : n/a
• references : [Trick, 2003]: A Dynamic Programming Approach for Consistency and Propagation
for Knapsack Constraints
Example:
CPModel m = new CPModel();
IntegerVariable[] items = new IntegerVariable[3];
items[0] = makeIntVar("item_1", 0, 5);
items[1] = makeIntVar("item_2", 0, 7);
items[2] = makeIntVar("item_3", 0, 10);
IntegerVariable sumWeight = makeIntVar("sumWeight", 0, 40, Options.V_BOUND);
IntegerVariable sumValue = makeIntVar("sumValue", 0, 34, Options.V_OBJECTIVE);
int[] weights = new int[]{7, 5, 3};
int[] values = new int[]{6, 4, 2};
Constraint knapsack = Choco.knapsackProblem(items, sumWeight, sumValue, weights, values)
;
m.addConstraint(knapsack);
Solver s = new CPSolver();
s.read(m);
s.maximize(true);
8.56
leq (constraint)
leq(x, y) states that x is less than or equal to y:
x≤y
• API :
– leq(IntegerExpressionVariable x, IntegerExpressionVariable y)
– leq(IntegerExpressionVariable x, int y)
– leq(int x, IntegerExpressionVariable y)
– leq(RealExpressionVariable x, RealExpressionVariable y)
– leq(RealExpressionVariable x, double y)
– leq(double x, RealExpressionVariable y)
• return type : Constraint
• options : n/a
• favorite domain : to complete.
• references :
global constraint catalog: leq
Example:
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8.57. LEQCARD (CONSTRAINT)
Model m = new CPModel();
Solver s = new CPSolver();
int c = 1;
IntegerVariable v = makeIntVar("v", 0, 2);
m.addConstraint(leq(v, c));
IntegerVariable v1 = makeIntVar("v1", 0, 2);
IntegerVariable v2 = makeIntVar("v2", 0, 2);
IntegerExpressionVariable w1 = plus(v1, 1);
IntegerExpressionVariable w2 = minus(v2, 1);
m.addConstraint(leq(w1, w2));
s.read(m);
s.solve();
8.57
leqCard (constraint)
leqCard(s, z) states that the cardinality of set s is less than or equal to z:
|s| ≤ z
• API :
– leqCard(SetVariable s, IntegerVariable z)
– leqCard(SetVariable s, int z)
• return type : Constraint
• options : n/a
• favorite domain : to complete
Example:
Model m = new CPModel();
Solver s = new CPSolver();
SetVariable set = makeSetVar("s", 1, 5);
IntegerVariable i = makeIntVar("card", 2, 3);
m.addConstraint(member(set, 3));
m.addConstraint(leqCard(set, i));
s.read(m);
s.solve();
8.58
lex (constraint)
lex(hx1 , .., xn i, hy1 , .., yn i) states a strict lexicographic ordering x <lex y:
∃ j = 1..n |
xj < yj
∧
xi = yi (∀ i < j)
• API : lex(IntegerVariable[] x, IntegerVariable[] y)
• return type : Constraint
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CHAPTER 8. CONSTRAINTS (MODEL)
• options :n/a
• favorite domain : to complete
• references :
– [Frisch et al., 2002]: Global Constraints for Lexicographic Orderings
– global constraint catalog: lex less
Example:
Model m = new CPModel();
Solver s = new CPSolver();
int n = 4;
int k = 2;
IntegerVariable[] vs1 = new IntegerVariable[n];
IntegerVariable[] vs2 = new IntegerVariable[n];
for (int i = 0; i < n; i++) {
vs1[i] = makeIntVar("" + i, 0, k);
vs2[i] = makeIntVar("" + i, 0, k);
}
m.addConstraint(lex(vs1, vs2));
s.read(m);
s.solve();
8.59
lexChain (constraint)
lexChain(hx11 , .., x1n i, . . . , hxp1 , .., xpn i) states a strict lexicographic ordering on a chain of p integer
vectors:
x1 <lex x2 <lex · · · <lex xp
• API : lexChain(IntegerVariable[]... x)
• return type : Constraint
• options : n/a
• favorite domain : to complete
• references :
– [Carlsson and Beldiceanu, 2002] Arc-Consistency for a chain of Lexicographic Ordering Constraints
– global constraint catalog: lex chain less
Example:
Model m = new CPModel();
Solver s = new CPSolver();
int n = 4;
int k = 2;
IntegerVariable[] vs1 = new
IntegerVariable[] vs2 = new
for (int i = 0; i < n; i++)
vs1[i] = makeIntVar("" +
vs2[i] = makeIntVar("" +
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IntegerVariable[n];
IntegerVariable[n];
{
i, 0, k);
i, 0, k);
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8.60. LEXCHAINEQ (CONSTRAINT)
}
m.addConstraint(lexChain(vs1, vs2));
s.read(m);
s.solve();
8.60
lexChainEq (constraint)
lexChainEq(hx11 , .., x1n i, . . . , hxp1 , .., xpn i) states a lexicographic ordering on a chain of p integer vectors:
x1 ≤lex x2 ≤lex · · · ≤lex xp
• API : lexChainEq(IntegerVariable[]... x)
• return type : Constraint
• options : n/a
• favorite domain : to complete
• references :
– [Carlsson and Beldiceanu, 2002] Arc-Consistency for a chain of Lexicographic Ordering Constraints
– global constraint catalog: lex chain lesseq
Example:
Model m = new CPModel();
Solver s = new CPSolver();
int n = 4;
int k = 2;
IntegerVariable[] vs1 = new IntegerVariable[n];
IntegerVariable[] vs2 = new IntegerVariable[n];
for (int i = 0; i < n; i++) {
vs1[i] = makeIntVar("" + i, 0, k);
vs2[i] = makeIntVar("" + i, 0, k);
}
m.addConstraint(lexChainEq(vs1, vs2));
s.read(m);
s.solve();
8.61
lexEq (constraint)
lexEq(hx1 , .., xn i, hy1 , .., yn i) states a lexicographic ordering x ≤lex y:
∃ j = 1..n |
xj ≤ yj
∧
xi = yi (∀ i < j)
• API : lexEq(IntegerVariable[] x, IntegerVariable[] y)
• return type : Constraint
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CHAPTER 8. CONSTRAINTS (MODEL)
• options :n/a
• favorite domain : to complete
• references :
– [Frisch et al., 2002]: Global Constraints for Lexicographic Orderings
– global constraint catalog: lex lesseq
Example: Example:
Model m = new CPModel();
Solver s = new CPSolver();
int n = 4;
int k = 2;
IntegerVariable[] vs1 = new IntegerVariable[n];
IntegerVariable[] vs2 = new IntegerVariable[n];
for (int i = 0; i < n; i++) {
vs1[i] = makeIntVar("" + i, 0, k);
vs2[i] = makeIntVar("" + i, 0, k);
}
m.addConstraint(lexEq(vs1, vs2));
s.read(m);
s.solve();
8.62
leximin (constraint)
check the specifications of the implemented version.
leximin(hx1 , .., xn i, hy1 , .., yn i) states a strict lexicographic ordering x0 <lex y 0 , where x0 and y 0 are
the permutations of x and y respectively sorted in increasing order.
sorting(hx1 , .., xn i, hx01 , .., x0n i) ∧ sorting(hy1 , .., yn i, hy10 , .., yn0 i) ∧ lex(hx01 , .., x0n i, hy10 , .., yn0 i)
• API :
– leximin(IntegerVariable[] x, IntegerVariable[] y)
– leximin(int[] x, IntegerVariable[] y)
• return type : Constraint
• options :n/a
• favorite domain : to complete
• references :
– [Frisch et al., 2003]: Multiset ordering constraints
– global constraint catalog: lex lesseq allperm (variant)
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable[] u = makeIntVarArray("u", 3, 2, 5);
IntegerVariable[] v = makeIntVarArray("v", 3, 2, 4);
m.addConstraint(leximin(u, v));
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8.63. LT (CONSTRAINT)
m.addConstraint(allDifferent(v));
s.read(m);
s.solve();
8.63
lt (constraint)
lt(x, y) states that x is strictly smaller than y:
x<y
• API :
– lt(IntegerExpressionVariable x, IntegerExpressionVariable y)
– lt(IntegerExpressionVariable x, int y)
– lt(int x, IntegerExpressionVariable y)
• return type : Constraint
• options : n/a
• favorite domain : to complete.
• references :
global constraint catalog: lt
Example:
Model m = new CPModel();
Solver s = new CPSolver();
int c = 1;
IntegerVariable v = makeIntVar("v", 0, 2);
m.addConstraint(lt(v, c));
s.read(m);
s.solve();
8.64
max (constraint)
8.64.1
max of a list
max(x, z) states that z is equal to the greater element of vector x:
z = max(x1 , x2 , ..., xn )
• API:
– max(IntegerVariable[] x, IntegerVariable z)
– max(IntegerVariable x1, IntegerVariable x2, IntegerVariable z)
– max(int x1, IntegerVariable x2, IntegerVariable z)
– max(IntegerVariable x1, int x2, IntegerVariable z)
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CHAPTER 8. CONSTRAINTS (MODEL)
• return type: Constraint
• options : n/a
• favorite domain : to complete
• references :
global constraint catalog: maximum
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable x = makeIntVar("x", 1, 5);
IntegerVariable y = makeIntVar("y", 1, 5);
IntegerVariable z = makeIntVar("z", 1, 5);
m.addVariables(Options.V_BOUND, x, y, z);
m.addConstraint(max(y, z, x));
s.read(m);
s.solve();
8.64.2
max of a set
max(s, x, z) states that z is equal to the greater element of vector x whose index is in set s:
z = max(xi )
i∈s
• API:
– max(SetVariable s, IntegerVariable[] x, IntegerVariable z)
• return type: Constraint
• options : n/a
• favorite domain : to complete
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable[] x = constantArray(new int[]{5, 7, 9, 10, 12, 3, 2});
IntegerVariable max = makeIntVar("max", 1, 100);
SetVariable set = makeSetVar("set", 0, x.length - 1);
m.addConstraints(max(set, x, max), leqCard(set, constant(5)));
s.read(m);
s.solve();
8.65
member (constraint)
member(x, s) states that integer x belongs to set s:
x∈s
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8.66. MIN (CONSTRAINT)
• API :
– member(int x, SetVariable s)
– member(SetVariable s, int x)
– member(SetVariable s, IntegerVariable x)
– member(IntegerVariable x, SetVariable s)
– member(member(SetVariable sv, IntegerVariable... vars)
– member(IntegerVariable x, int[] s)
– member(IntegerVariable x, int lower, int upper)
• return type : Constraint
• options :n/a
• favorite domain : to complete
• references :
global constraint catalog: in set
Examples: 1. using a set variable
Model m = new CPModel();
Solver s = new CPSolver();
int x = 3;
int card = 2;
SetVariable y = makeSetVar("y", 2, 4);
m.addConstraint(member(y, x));
m.addConstraint(eqCard(y, card));
s.read(m);
s.solveAll();
2. using an array of integers
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable var = makeIntVar("v1", 0, 100, Options.V_BOUND);
int[] values = new int[]{0, 25, 50, 75, 100};
m.addConstraint(member(var, values));
s.read(m);
s.solve();
8.66
min (constraint)
8.66.1
min of a list
min(x, z) states that z is equal to the smaller element of vector x:
z = min(x1 , x2 , ..., xn ).
• API:
– min(IntegerVariable[] x, IntegerVariable z)
– min(IntegerVariable x1, IntegerVariable x2, IntegerVariable z)
– min(int x1, IntegerVariable x2, IntegerVariable z)
– min(IntegerVariable x1, int x2, IntegerVariable z)
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CHAPTER 8. CONSTRAINTS (MODEL)
• return type: Constraint
• options : n/a
• favorite domain : to complete
• references :
global constraint catalog: minimum
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable x = makeIntVar("x", 1, 5);
IntegerVariable y = makeIntVar("y", 1, 5);
IntegerVariable z = makeIntVar("z", 1, 5);
m.addVariables(Options.V_BOUND, x, y, z);
m.addConstraint(min(y, z, x));
s.read(m);
s.solve();
8.66.2
min of a set
min(s, x, z) states that z is equal to the smaller element of vector x whose index is in set s:
z = min(xi ).
i∈s
• API:
– min(SetVariable s,IntegerVariable[] x, IntegerVariable z)
• return type: Constraint
• options : n/a
• favorite domain : to complete
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable[] x = constantArray(new int[]{5, 7, 9, 10, 12, 3, 2});
IntegerVariable min = makeIntVar("min", 1, 100);
SetVariable set = makeSetVar("set", 0, x.length - 1);
m.addConstraints(min(set, x, min), leqCard(set, constant(5)));
s.read(m);
s.solve();
8.67
mod (constraint)
mod(x1 , x2 , x3 ) states that x1 is congruent to x2 modulo x3 :
x1 ≡ x2
mod x3
• API : mod(IntegerVariable x1, IntegerVariable x2, int x3)
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8.68. MULTICOSTREGULAR (CONSTRAINT)
• return type : Constraint
• options : n/a
• favorite domain : n/a
Example:
Model m = new CPModel();
Solver s = new CPSolver();
int n = 4;
int k = 2;
IntegerVariable[] vs1 = new IntegerVariable[n];
IntegerVariable[] vs2 = new IntegerVariable[n];
for (int i = 0; i < n; i++) {
vs1[i] = makeIntVar("" + i, 0, k);
vs2[i] = makeIntVar("" + i, 0, k);
}
m.addConstraint(lex(vs1, vs2));
s.read(m);
s.solve();
8.68
multiCostRegular (constraint)
multiCostRegular(hz1 , .., zp i, hx1 , .., xn i, L(Π), hci,j,k i) states that sequence hx1 , .., xn i is a word
belonging to the regular language L(Π) and that each zk is its cost computed as the sum of the
individual symbol weights ci,xi ,k :
hx1 , .., xn i ∈ L(Π)
∧
n
X
ci,xi ,k = zk , ∀k = 1..p.
i=1
multiCostRegular models the conjunction of p costRegular constraints, or the conjunction of a
regular constraint with p assignment cost functions. Like them, it is useful for modelling sequencing
rules in personnel scheduling and rostering problems. Furthermore it allows to handle together several
linear counters and costs on the sequence x. For example, one may count all the value occurrences
like with a globalCardinality constraint. Counters can also model assignment costs in optimization
problems or violation costs in over-constrained problems. For example, counting the occurrence number
of a pattern allows to determine the violation cost of a soft forbidden pattern rule.
The filtering algorithm [Menana and Demassey, 2009] of multiCostRegular does not achieve GAC
(as it would be NP-hard) but it dominates the decompositions in regular or costRegular constraints.
The accepting language is specified by a deterministic finite automaton (DFA) Π encoded as an
object of class Automaton (see costRegular for a short API). The cost functions are vectors of weights
on the transitions of Π. They are encoded as one matrix int c[n][m][p][pi.getNbStates()] such
that c[i][j][k][s] is the cost of assigning variable xi to value j at state s on dimension k. When the
transition costs are independent of their initial states, a second API allows to specify a cost matrix int
c[n][m][p].
• API :
– multiCostRegular(IntegerVariable[] z, IntegerVariable[] x, FiniteAutomaton pi, int[][][]
c)
– multiCostRegular(IntegerVariable[] z, IntegerVariable[] x, FiniteAutomaton pi, int[][][][]
c)
• return type : Constraint
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CHAPTER 8. CONSTRAINTS (MODEL)
• options : n/a
• favorite domain : to complete
• references :
[Menana and Demassey, 2009]: Sequencing and Counting with the multicost-regular Constraint
Example:
import choco.kernel.model.constraints.automaton.FA.Automaton;
Model m = new CPModel();
int
int
int
int
nTime = 14; //
nAct = 3; // 3
DAY = 0, NIGHT
nCounters = 4;
2 weeks: 14 days
shift types:
= 1, REST = 2;
// cost (0), #DAY (1), #NIGHT (2), #WORK (3)
IntegerVariable[] x = makeIntVarArray("x", nTime, 0, nAct - 1, Options.V_ENUM);
IntegerVariable[] z = new IntegerVariable[4];
z[0] = makeIntVar("z", 30, 800, Options.V_BOUND);
z[1] = makeIntVar("D", 0, 7, Options.V_BOUND); //
z[2] = makeIntVar("N", 3, 7, Options.V_BOUND); //
z[3] = makeIntVar("W", 7, 9, Options.V_BOUND); //
// 30 <= cost <= 80
0 <= #DAY <= 7
3 <= #NIGHT <= 7
7 <= #WORK <= 9
FiniteAutomaton auto = new FiniteAutomaton();
int start = auto.addState();
auto.setInitialState(start);
auto.setFinal(start);
int first = auto.addState();
auto.addTransition(start, first, DAY); // transition (0,D,1)
int second = auto.addState();
auto.addTransition(first, second, DAY, NIGHT); // transitions (1,D,2), (1,N,2)
auto.addTransition(second, start, REST); // transition (2,R,0)
auto.addTransition(start, second, NIGHT); // transition (0,N,2)
int[][][][] c = new int[nTime][nAct][nCounters][auto.getNbStates()];
for (int i = 0; i < c.length; i++) {
c[i][DAY][0] = new int[]{3, 5, 0};
c[i][DAY][1] = new int[]{1, 1, 0};
c[i][DAY][3] = new int[]{1, 1, 0};
c[i][NIGHT][0] = new int[]{8, 9, 0};
c[i][NIGHT][2] = new int[]{1, 1, 0};
c[i][NIGHT][3] = new int[]{1, 1, 0};
c[i][REST][0] = new int[]{0, 0, 2};
}
ICostAutomaton cauto = CostAutomaton.makeMultiResources(auto,c,z);
m.addConstraint(multiCostRegular(z, x, cauto));
Solver s = new CPSolver();
s.read(m);
System.out.println(s.minimize(s.getVar(z[0]),false));
// s.solve();
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8.69. NAND (CONSTRAINT)
System.out.println(s.getVar(z[0]).pretty());
System.out.println(s.runtimeStatistics());
8.69
nand (constraint)
nand(hC1 , .., Cn i) states that at least one constraint in arguments is not satisfied:
C1 ↑ C2 ↑ . . . ↑ Cn
nand(hb1 , .., bn i) states that at least one boolean in arguments is false:
(b1 = 1) ↑ (b2 = 1) ↑ . . . ↑ (bn = 1)
• API :
– nand(Constraint... c)
– nand(IntegerVariable... b)
• return type : Constraint
• options : n/a
• favorite domain : n/a
• references :
global constraint catalog: nand
Examples:
• example1:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable v1 = makeIntVar("v1", 0, 1);
IntegerVariable v2 = makeIntVar("v2", 0, 1);
m.addConstraint(nand(eq(v1, 1), eq(v2, 1)));
s.read(m);
s.solve();
• example2
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable[] vars = makeBooleanVarArray("b", 10);
m.addConstraint(and(vars));
s.read(m);
s.solve();
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CHAPTER 8. CONSTRAINTS (MODEL)
8.70
neq (constraint)
neq states that the two arguments are different:
x 6= y.
• API :
– neq(IntegerExpressionVariable x, IntegerExpressionVariable y)
– neq(IntegerExpressionVariable x, int y)
– neq(int x, IntegerExpressionVariable y)
• return type : Constraint
• options : n/a
• favorite domain : to complete.
• references :
global constraint catalog: neq
Examples:
• example1:
Model m = new CPModel();
Solver s = new CPSolver();
int c = 1;
IntegerVariable v = makeIntVar("v", 0, 2);
m.addConstraint(neq(v, c));
s.read(m);
s.solve();
• example2
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable v1 = makeIntVar("v1", 0, 2);
IntegerVariable v2 = makeIntVar("v2", 0, 2);
IntegerExpressionVariable w1 = plus(v1, 1);
IntegerExpressionVariable w2 = minus(v2, 1);
m.addConstraint(neq(w1, w2));
s.read(m);
s.solve();
8.71
neqCard (constraint)
neqCard(s, z) states that the cardinality of set s is not equal to z:
|s| =
6 z
CHOCO solver documentation
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8.72. NOT (CONSTRAINT)
• API :
– neqCard(SetVariable s, IntegerVariable z)
– neqCard(SetVariable s, int z)
• return type : Constraint
• options : n/a
• favorite domain : to complete
Example:
Model m = new CPModel();
Solver s = new CPSolver();
SetVariable set = makeSetVar("s", 1, 5);
IntegerVariable card = makeIntVar("card", 2, 3);
m.addConstraint(member(set, 3));
m.addConstraint(neqCard(set, card));
s.read(m);
s.solve();
8.72
not (constraint)
not(c) holds if and only if constraint c does not hold:
¬c
• API : not(Constraint c)
• return type : Constraint
• options : n/a
• favorite domain : n/a
Example :
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable x = makeIntVar("x", 1, 10);
m.addConstraint(not(geq(x, 3)));
s.read(m);
s.solve();
8.73
notMember (constraint)
notMember(x, s) states that integer x does not belong to s:
x 6∈ s
• API :
CHOCO solver documentation
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CHAPTER 8. CONSTRAINTS (MODEL)
– notMember(int x, SetVariable s)
– notMember(SetVariable s, int x)
– notMember(SetVariable s, IntegerVariable x)
– notMember(IntegerVariable x, SetVariable s)
– notMember(IntegerVariable x, int[] s)
– notMember(IntegerVariable x, int lower, int upper)
• return type : Constraint
• options :n/a
• favorite domain : to complete
Example:
• with a set variable
Model m = new CPModel();
Solver s = new CPSolver();
int x = 3;
int card = 2;
SetVariable y = makeSetVar("y", 2, 4);
m.addConstraint(notMember(y, x));
m.addConstraint(eqCard(y, card));
s.read(m);
s.solveAll();
• with a collection of values
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable var = makeIntVar("v1", 0, 100, Options.V_BOUND);
int[] values = new int[]{10, 20, 30, 40, 50, 60, 70, 80, 90};
m.addConstraint(notMember(var, values));
s.read(m);
s.solve();
8.74
nor (constraint)
nor(c1 , . . . , cn ) states that at least one constraint in arguments is satisfied:
c1 ↓ c2 ↓ . . . ↓ cn
nor(b1 , . . . , bn ) states that at least one boolean variable in argument is true:
(b1 = 1) ↓ (b2 = 1) ↓ . . . ↓ (bn = 1)
• API :
– nor(Constraint... c)
– nor(IntegerVariable... b)
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8.75. NTH (CONSTRAINT)
• return type : Constraint
• options : n/a
• favorite domain : n/a
• references :
global constraint catalog: nor
Examples:
• example1:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable v1 = makeIntVar("v1", 0, 1);
IntegerVariable v2 = makeIntVar("v2", 0, 1);
m.addConstraint(nor(eq(v1, 1), eq(v2, 1)));
s.read(m);
s.solve();
• example2
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable[] vars = makeBooleanVarArray("b", 10);
m.addConstraint(or(vars));
s.read(m);
s.solve();
8.75
nth (constraint)
nth is the well known element constraint. Several APIs are available:
• nth(i, hx0 , .., xn i, y) states that y = xi
• nth(i, hx0 , .., xn i, y, o) ensures that y = xi−o (o is an offset for shifting values)
• nth(i, j, hxi,j i, y) ensures that y = xi,j
• API :
– nth(IntegerVariable i, int[] x, IntegerVariable y)
– nth(String option, IntegerVariable i, int[] x, IntegerVariable y)
– nth(IntegerVariable i, IntegerVariable[] x, IntegerVariable y)
– nth(IntegerVariable i, int[] x, IntegerVariable y, int offset)
– nth(String option, IntegerVariable i, int[] x, IntegerVariable y, int offset)
– nth(IntegerVariable i, IntegerVariable[] x, IntegerVariable y, int offset)
– nth(String option, IntegerVariable i, IntegerVariable[] x, IntegerVariable y, int offset
)
– nth(IntegerVariable i, IntegerVariable j, int[][] x, IntegerVariable y)
• return type : Constraint
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CHAPTER 8. CONSTRAINTS (MODEL)
• options :
– no option
– Options.C NTH G for global consistency
• favorite domain : to complete
• references :
global constraint catalog: element
Example:
Model m = new CPModel();
Solver s = new CPSolver();
int[][] values = new int[][]{
{1, 2, 0, 4, -323},
{2, 1, 0, 3, 42},
{6, 1, -7, 4, -40},
{-1, 0, 6, 2, -33},
{2, 3, 0, -1, 49}};
IntegerVariable index1 = makeIntVar("index1", -3, 10);
IntegerVariable index2 = makeIntVar("index2", -3, 10);
IntegerVariable var = makeIntVar("value", -20, 20);
m.addConstraint(nth(index1, index2, values, var));
s.read(m);
s.solveAll();
8.76
occurrence (constraint)
occurrence(v, z, x) states that z is equal to the number of elements in x with value v:
z = |{i | xi = v}|
This is a specialization of globalCardinality with only one value counter, and of among with exactly
one counted value.
• API:
– occurrence(IntegerVariable z, IntegerVariable[] x, int v)
– occurrence(int z, IntegerVariable[] x, int v)
– occurrence(int v, IntegerVariable z, IntegerVariable... x)
• return type : Constraint
• options :n/a
• favorite domain : to complete
• references :
global constraint catalog: count
Example:
Model m = new CPModel();
Solver s = new CPSolver();
int n=7;
IntegerVariable[] x = makeIntVarArray("X", n, 0, 10);
IntegerVariable z = makeIntVar("Z", 0, 10);
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8.77. OCCURRENCEMAX (CONSTRAINT)
m.addConstraint(occurrence(z, x, 3));
s.read(m);
s.solve();
8.77
occurrenceMax (constraint)
occurrenceMax(v, z, x) states that z is at least equal to the number of elements in x with value v:
z ≥ |{i | xi = v}|
See also occurrence.
• API:
– occurrenceMax(IntegerVariable z, IntegerVariable[] x, int v)
– occurrenceMax(int v, IntegerVariable z, IntegerVariable... x)
• return type : Constraint
• options :n/a
• favorite domain : to complete
• references :
global constraint catalog: count
Example:
Model m = new CPModel();
Solver s = new CPSolver();
int n=7;
IntegerVariable[] x = makeIntVarArray("X", n, 0, 10);
IntegerVariable z = makeIntVar("Z", 0, 10);
m.addConstraint(occurrenceMax(z, x, 3));
s.read(m);
s.solve();
8.78
occurrenceMin (constraint)
occurrenceMin(v, z, x) states that z is at most equal to the number of elements in x with value v:
z ≤ |{i | xi = v}|
See also occurrence.
• API:
– occurrenceMin(IntegerVariable z, IntegerVariable[] x, int v)
– occurrenceMin(int v, IntegerVariable z, IntegerVariable... x)
• return type : Constraint
• options :n/a
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CHAPTER 8. CONSTRAINTS (MODEL)
• favorite domain : to complete
• references :
global constraint catalog: count
Example:
Model m = new CPModel();
Solver s = new CPSolver();
int n=7;
IntegerVariable[] x = makeIntVarArray("X", n, 0, 10);
IntegerVariable z = makeIntVar("Z", 0, 10);
m.addConstraint(occurrenceMin(z, x, 3));
s.read(m);
s.solve();
8.79
oppositeSign (constraint)
oppositeSign(x, y) states that the two arguments have opposite signs:
xy < 0
Note that 0 has both signs then constraint fails if x or y is equal to 0.
• API : oppositeSign(IntegerExpressionVariable x, IntegerExpressionVariable y)
• return type : Constraint
• options :n/a
• favorite domain : to complete
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable x = makeIntVar("x", -1, 1);
IntegerVariable y = makeIntVar("y", -1, 1);
IntegerVariable z = makeIntVar("z", 0, 1000);
m.addConstraint(oppositeSign(x,y));
m.addConstraint(eq(z, plus(mult(x, -425), mult(y, 391))));
s.read(m);
s.solve();
8.80
or (constraint)
or(c1 , . . . , cn ) states that at least one constraint in arguments is satisfied:
c1 ∨ c2 ∨ . . . ∨ cn
or(b1 , . . . , bn ) states that at least one boolean variable in argument is true:
(b1 = 1) ∨ (b2 = 1) ∨ . . . ∨ (bn = 1)
CHOCO solver documentation
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8.81. PACK (CONSTRAINT)
• API :
– or(Constraint... c)
– or(IntegerVariable... b)
• return type : Constraint
• options : n/a
• favorite domain : n/a
• references :
global constraint catalog: or
Examples:
• example1:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable v1 = makeIntVar("v1", 0, 1);
IntegerVariable v2 = makeIntVar("v2", 0, 1);
m.addConstraint(or(eq(v1, 1), eq(v2, 1)));
s.read(m);
s.solve();
• example2
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable[] vars = makeBooleanVarArray("b", 10);
m.addConstraint(or(vars));
s.read(m);
s.solve();
8.81
pack (constraint)
pack(items, load, bin, size) states that a collection of items is packed into different bins, such
that the total size of the items in each bin does not exceed the bin capacity:
X
load[b] =
size[i], ∀ bin b
i∈items[b]
i ∈ items[b] ⇐⇒ bin[i] = b,
∀ bin b, ∀ item i
pack is a bin packing constraint based on [Shaw, 2004].
• API :
– pack(SetVariable[] items, IntegerVariable[] load, IntegerVariable[] bin, IntegerConstantVariable
[] size, String... options)
– pack(PackModeler modeler,String... options): PackModeler is a high-level modeling object.
– pack(int[] sizes, int nbBins, int capacity, String... options): build instance with PackModeler.
• Variables:
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CHAPTER 8. CONSTRAINTS (MODEL)
– SetVariable[] items:
items[b] is the set of items packed into bin b.
– IntegerVariable[] load:
– IntegerVariable[] bin:
load[b] is the total size of the items packed into bin b.
bin[i] is the bin where item i is packed into.
– IntegerConstantVariable[] size:
size[i] is the size of item i.
• return type : Constraint
• options :
– SettingType.ADDITIONAL RULES.getOptionName(): additional filtering rules recommended
– SettingType.DYNAMIC LB.getOptionName(): feasibility tests based on dynamic lower bounds
for 1D-bin packing
– SettingType.FILL BIN.getOptionName(): dominance rule: fill a bin when an item fit into
pertfectly equal-sized items and bins must be equivalent
– SettingType.LAST BINS EMPTY.getOptionName(): empty bins are the last ones
• favorite domain : to complete
• references :
– [Shaw, 2004]: A constraint for bin packing
– global constraint catalog: bin packing (variant)
Example:
Take a look at samples.pack to see advanced use of the constraint.
import choco.cp.solver.SettingType;
import choco.cp.solver.search.integer.varselector.StaticVarOrder;
Model m = new CPModel();
PackModel pm = new PackModel(new int[]{5,3,2,6,8,5},5,10);
m.addConstraint(pack(pm, Options.C_PACK_AR));
Solver s = new CPSolver();
s.read(m);
s.solve();
8.82
precedenceReified (constraint)
precedenceReified(x1 , d, x2 , b) states that x1 plus duration d is less than or equal to x2 requires
boolean b to be true:
b
⇐⇒
x1 + d ≤ x2
• API : precedenceReified(IntegerVariable x1, int d, IntegerVariable x2, IntegerVariable b)
• return type : Constraint
• options :n/a
• favorite domain : to complete
Example:
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8.83. PRECEDENCEIMPLIED (CONSTRAINT)
int k1 = 5;
Model m = new CPModel();
IntegerVariable x = makeIntVar("x", 1, 10);
IntegerVariable y = makeIntVar("y", 1, 10);
m.addVariables(Options.V_BOUND, x, y);
IntegerVariable z = makeIntVar("z", 0, 1);
m.addConstraint(precedenceReified(x,k1,y,z));
Solver s = new CPSolver();
s.read(m);
s.solve();
8.83
precedenceimplied (constraint)
To complete
8.84
precedence (constraint)
To complete
8.85
precedencedisjoint (constraint)
To complete
8.86
regular (constraint)
regular(x, L(Π)) states that sequence x is a word belonging to the regular language L(Π):
(x1 , . . . , xn ) ∈ L(Π)
The accepting language can be specified either by a deterministic finite automaton (DFA), a list of
feasible or infeasible tuples, or a regular expression:
DFA: Automaton Π is defined on a given alphabet Σ ⊆ Z by a set Q = {0, . . . , m} of states, a subset
A ⊆ Q of final or accepting states and a table ∆ ⊆ Q×Σ× Q of transitions between states. ∆ is
encoded as List<Transition> where a Transition object δ = new Transition(qi , σ, qj ) is made of
three integers expressing the ingoing state qi , the label σ, and the outgoing state qj . Automaton Π
is a DFA if ∆ is finite and if it has only one initial state (here, state 0 is considered as the unique
initial state) and no two transitions sharing the same ingoing state and the same label.
FiniteAutomaton: is another API for building a DFA (manually, or from a regular expression, or
from a dk.brics.Automaton) and operating on them (intersection, union, complement) in a more
flexible way. Using this API leads to another implementation of the constraint: FastRegular. See
costRegular for a short API of FiniteAutomaton.
feasible tuples: regular can be used as an extensional constraint. Given the list of feasible tuples for
sequence x, this API builds a DFA from the list, and then enforces GAC on the constraint. Using
regular can be more efficient than a standard GAC algorithm on tables of tuples if the tuples
are structured so that the resulting DFA is compact. The DFA is built from the list of tuples by
computing incrementally the minimal DFA after each addition of tuple.
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CHAPTER 8. CONSTRAINTS (MODEL)
infeasible tuples: An another API allows to specify the list of infeasible tuples and then builds the
corresponding feasible DFA. This operation requires to know the entire alphabet, hence this API
has two mandatory table fields min and max defining the minimum and maximum values of each
variable xi .
regular expression: Finally, the regular constraint can be based on a regular expression, such as
String regexp = "(1|2)3{4}5*";. This expression recognizes any sequences starting by one 1 or
one 2, then four consecutive 3 followed by any (possibly empty) sequences of 5.
Warning ! DFA and FiniteAutomaton are both based on the dk.brics library. The construction of
these objects is non-deterministic and the order the filtering occur (not the result) may vary at each
execution. This may results in different first solutions when branching dynamically using weighted
degrees-base heuristics for example.
• API :
– regular(IntegerVariable[] x, FiniteAutomaton pi)
– regular(IntegerVariable[] x, DFA pi)
– regular(IntegerVariable[] x, List<int[]> feasTuples)
– regular(IntegerVariable[] x, List<int[]> infeasTuples, int[] min, int[] max)
– regular(IntegerVariable[] x, String regexp)
• return type : Constraint
• options :n/a
• favorite domain : to complete
• references :
[Pesant, 2004]: A regular language membership constraint
Examples:
• example with FiniteAutomaton: see costRegular.
• example 1 with DFA:
import choco.kernel.model.constraints.automaton.DFA;
import choco.kernel.model.constraints.automaton.Transition;
import choco.kernel.model.constraints.Constraint;
//1- Create the model
Model m = new CPModel();
int n = 6;
IntegerVariable[] vars = new IntegerVariable[n];
for (int i = 0; i < vars.length; i++) {
vars[i] = makeIntVar("v" + i, 0, 5);
}
//2- Build the list of transitions of the DFA
List<Transition> t = new LinkedList<Transition>();
t.add(new Transition(0, 1, 1));
t.add(new Transition(1, 1, 2));
// transition with label 1 from state 2 to state 3
t.add(new Transition(2, 1, 3));
t.add(new Transition(3, 3, 0));
t.add(new Transition(0, 3, 0));
//3- Two final states: 0, 3
List<Integer> fs = new LinkedList<Integer>();
fs.add(0); fs.add(3);
//4- Build the DFA
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8.86. REGULAR (CONSTRAINT)
DFA auto = new DFA(t, fs, n);
//5- add the constraint
m.addConstraint(regular(vars, auto));
//6- create the solver, read the model and solve it
Solver s = new CPSolver();
s.read(m);
s.solve();
do {
for (int i = 0; i < n; i++)
System.out.print(s.getVar(vars[i]).getVal());
System.out.println("");
} while (s.nextSolution());
//7- Print the number of solution found
System.out.println("Nb_sol : " + s.getNbSolutions());
• example 2 with feasible tuples:
//1- Create the model
Model m = new CPModel();
IntegerVariable v1 = makeIntVar("v1", 1, 4);
IntegerVariable v2 = makeIntVar("v2", 1, 4);
IntegerVariable v3 = makeIntVar("v3", 1, 4);
//2- add some allowed tuples (here, the tuples define a all_equal constraint)
List<int[]> tuples = new LinkedList<int[]>();
tuples.add(new int[]{1, 1, 1});
tuples.add(new int[]{2, 2, 2});
tuples.add(new int[]{3, 3, 3});
tuples.add(new int[]{4, 4, 4});
//3- add the constraint
m.addConstraint(regular(new IntegerVariable[]{v1, v2, v3}, tuples));
//4- Create the solver, read the model and solve it
Solver s = new CPSolver();
s.read(m);
s.solve();
do {
System.out.println("("+s.getVar(v1)+","+s.getVar(v2)+","+s.getVar(v3)+")");
} while (s.nextSolution());
//5- Print the number of solution found
System.out.println("Nb_sol : " + s.getNbSolutions());
• example 3 with regular expression:
//1- Create the model
Model m = new CPModel();
int n = 6;
IntegerVariable[] vars = makeIntVarArray("v", n, 0, 5);
//2- add the constraint
String regexp = "(1|2)(3*)(4|5)";
m.addConstraint(regular(vars, regexp));
//3- Create the solver, read the model and solve it
Solver s = new CPSolver();
s.read(m);
s.solve();
do {
for (int i = 0; i < n; i++)
System.out.print(s.getVar(vars[i]).getVal());
System.out.println("");
} while (s.nextSolution());
//4- Print the number of solution found
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CHAPTER 8. CONSTRAINTS (MODEL)
System.out.println("Nb_sol : " + s.getNbSolutions());
8.87
reifiedAnd (constraint)
To complete
8.88
reifiedConstraint (constraint)
• reifiedConstraint(b, C) states that boolean b is true if and only if constraint C holds:
(b = 1) ⇐⇒ C
• reifiedConstraint(b, C1 , C2 ) states that boolean b is true if and only if C1 holds, and b is
false if and only if C2 holds (C2 must be the negation of constraint of C1 ):
(b ∧ C1 ) ∨ (¬b ∧ C2 )
• API :
– reifiedConstraint(IntegerVariable b, Constraint c)
– reifiedConstraint(IntegerVariable b, Constraint c1, Constraint c2)
• return type : Constraint
• options : n/a
• favorite domain : n/a
The constraint C to reify has to provide its negation ¬C (the negation is needed for propagation).
Most basic constraints of Choco provides their negation by default, and can then be reified using the first
API. The second API attends to reify user-defined constraints as it allows the user to directly specify
the negation constraint.
The reifiedConstraint filter algorithm:
1. if b is instantiated to 1 (resp. to 0), then C (resp. ¬C) is propagated
2. otherwise
(a) if C is entailed, b is set to 1
(b) else if C is failed, b is set to 0.
Example:
CPModel m = new CPModel();
CPSolver s = new CPSolver();
IntegerVariable b = makeIntVar("b", 0, 1);
IntegerVariable x = makeIntVar("x", 0, 10);
IntegerVariable y = makeIntVar("y", 0, 10);
// reified constraint (x<=y)
m.addConstraint(reifiedConstraint(b, leq(x, y)));
s.read(m);
s.solveAll();
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8.89. REIFIEDLEFTIMP (CONSTRAINT)
8.89
reifiedLeftImp (constraint)
To complete
8.90
reifiedNot (constraint)
To complete
8.91
reifiedOr (constraint)
To complete
8.92
reifiedRightImp (constraint)
To complete
8.93
reifiedXnor (constraint)
To complete
8.94
reifiedXor (constraint)
To complete
8.95
relationPairAC (constraint)
relationPairAC(x, y, rel) states an extensional binary constraint on (x, y) defined by the binary
relation rel:
(x, y) ∈ rel
Many constraints of the same kind often appear in a model. Relations can therefore often be shared
among many constraints to spare memory.
The API is duplicated to allow definition of options.
• API :
– relationPairAC(IntegerVariable x, IntegerVariable y, BinRelation rel)
– relationPairAC(String options, IntegerVariable x, IntegerVariable y, BinRelation rel)
• return type : Constraint
• options :
– no option : use AC3 (default arc consistency)
– Options.C EXT AC3: to get AC3 algorithm (searching from scratch for supports on all values)
– Options.C EXT AC2001: to get AC2001 algorithm (maintaining the current support of each
value)
– Options.C EXT AC32: to get AC3rm algorithm (maintaining the current support of each value
in a non backtrackable way)
– Options.C EXT AC322: to get AC3 with the used of BitSet to know if a support still exists
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CHAPTER 8. CONSTRAINTS (MODEL)
• favorite domain : to complete
Example:
import choco.kernel.solver.constraints.integer.extension.CouplesTest;
import choco.kernel.solver.constraints.integer.extension.TuplesTest;
public static class MyEquality extends CouplesTest {
public boolean checkCouple(int x, int y) {
return x == y;
}
}
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable v1 = makeIntVar("v1", 1, 4);
IntegerVariable v2 = makeIntVar("v2", 1, 4);
IntegerVariable v3 = makeIntVar("v3", 3, 6);
m.addConstraint(relationPairAC(Options.C_EXT_AC32, v1, v2, new MyEquality()));
m.addConstraint(relationPairAC(Options.C_EXT_AC32, v2, v3, new MyEquality()));
s.read(m);
s.solveAll();
8.96
relationTupleAC (constraint)
relationTupleAC(x, rel) states an extensional constraint on (x1 , . . . , xn ) defined by the n-ary relation rel, and then enforces arc consistency:
(x1 , . . . , xn ) ∈ rel
Many constraints of the same kind often appear in a model. Relations can therefore often be shared
among many constraints to spare memory. The API is duplicated to define options.
• API:
– relationTupleAC(IntegerVariable[] x, LargeRelation rel)
– relationTupleAC(String options, IntegerVariable[] x, LargeRelation rel)
• return type: Constraint
• options :
– no option: use AC32 (default arc consistency)
– Options.C EXT AC32: to get AC3rm algorithm (maintaining the current support of each value
in a non backtrackable way)
– Options.C EXT AC2001: to get AC2001 algorithm (maintaining the current support of each
value)
– Options.C EXT AC2008: to get AC2008 algorithm (maintained by STR)
• favorite domain : to complete
Example :
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8.97. RELATIONTUPLEFC (CONSTRAINT)
public static class NotAllEqual extends TuplesTest {
public boolean checkTuple(int[] tuple) {
for (int i = 1; i < tuple.length; i++) {
if (tuple[i - 1] != tuple[i]) return true;
}
return false;
}
}
Model m = new CPModel();
IntegerVariable x = makeIntVar("x", 1, 5);
IntegerVariable y = makeIntVar("y", 1, 5);
IntegerVariable z = makeIntVar("z", 1, 5);
m.addConstraint(relationTupleAC(new IntegerVariable[]{x, y, z}, new NotAllEqual()));
Solver s = new CPSolver();
s.read(m);
s.solveAll();
8.97
relationTupleFC (constraint)
relationTupleFC(x, rel) states an extensional constraint on (x1 , . . . , xn ) defined by the n-ary relation rel, and then enforces forward checking:
(x1 , . . . , xn ) ∈ rel
Many constraints of the same kind often appear in a model. Relations can therefore often be shared
among many constraints to spare memory.
• API: relationTupleFC(IntegerVariable[] x, LargeRelation rel)
• return type: Constraint
• options : n/a
• favorite domain : to complete
Example :
public static class NotAllEqual extends TuplesTest {
public boolean checkTuple(int[] tuple) {
for (int i = 1; i < tuple.length; i++) {
if (tuple[i - 1] != tuple[i]) return true;
}
return false;
}
}
Model m = new CPModel();
IntegerVariable x = makeIntVar("x", 1, 5);
IntegerVariable y = makeIntVar("y", 1, 5);
IntegerVariable z = makeIntVar("z", 1, 5);
m.addConstraint(relationTupleFC(new IntegerVariable[]{x, y, z}, new NotAllEqual()));
Solver s = new CPSolver();
s.read(m);
s.solveAll();
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CHAPTER 8. CONSTRAINTS (MODEL)
8.98
sameSign (constraint)
sameSign(x, y) states that the two arguments have the same sign:
xy ≥ 0
Note that 0 has both signs then constraint holds if x or y is equal to 0.
• API : sameSign(IntegerExpressionVariable x, IntegerExpressionVariable y)
• return type : Constraint
• options :n/a
• favorite domain : to complete
Example:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable x = makeIntVar("x", -1, 1);
IntegerVariable y = makeIntVar("y", -1, 1);
IntegerVariable z = makeIntVar("z", 0, 1000);
m.addConstraint(oppositeSign(x,y));
m.addConstraint(eq(z, plus(mult(x, -425), mult(y, 391))));
s.read(m);
s.solve();
System.out.println(s.getVar(z).getVal());
8.99
setDisjoint (constraint)
setDisjoint(s1 , . . . , sn ) states that the arguments are pairwise disjoint:
si ∩ sj = ∅,
∀ i 6= j
• API : setDisjoint(SetVariable[] sv)
• return type : Constraint
• options :n/a
• favorite domain : to complete
Example:
Model m = new CPModel();
Solver s = new CPSolver();
SetVariable x = makeSetVar("X", 1, 3);
SetVariable y = makeSetVar("Y", 1, 3);
SetVariable z = makeSetVar("Z", 1, 3);
Constraint c1 = setDisjoint(x, y, z);
m.addConstraint(c1);
s.read(m);
s.solveAll();
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8.100. SETINTER (CONSTRAINT)
8.100
setInter (constraint)
setInter(s1 , s2 , s3 ) states that the third set s3 is exactly the intersection of the two first sets:
s1 ∩ s2 = s3
• API : setInter(SetVariable s1, SetVariable s2, SetVariable s3)
• return type : Constraint
• options :n/a
• favorite domain : to complete
Example:
Model m = new CPModel();
Solver s = new CPSolver();
SetVariable x = makeSetVar("X", 1, 3);
SetVariable y = makeSetVar("Y", 1, 3);
SetVariable z = makeSetVar("Z", 2, 3);
Constraint c1 = setInter(x, y, z);
m.addConstraint(c1);
s.read(m);
s.solveAll();
8.101
setUnion (constraint)
setUnion(sv, sunion ) states that the sunion set is exactly the union of the sets sv:
sv1 ∪ sv2 ∪ . . . svi ∪ svi+1 . . . ∪ svn = sunion
• API :
– setUnion(SetVariable s1, SetVariable s2, SetVariable union)
– setUnion(SetVariable[] sv, SetVariable union)
• return type : Constraint
• options :n/a
• favorite domain : to complete
Example:
Model m = new CPModel();
Solver s = new CPSolver();
SetVariable x = makeSetVar("X", 1, 3);
SetVariable y = makeSetVar("Y", 3, 5);
SetVariable z = makeSetVar("Z", 0, 6);
Constraint c1 = setUnion(x, y, z);
m.addConstraint(c1);
s.read(m);
s.solveAll();
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CHAPTER 8. CONSTRAINTS (MODEL)
8.102
sorting (constraint)
sorting(x, y) holds on the set of variables being either in x or in y, and is satisfied by v if and only
if v(y) is the sorted version of v(x) in increasing order.
y = xs orted
• API: sorting(IntegerVariable[] x, IntegerVariable[] y)
• return type : Constraint
• options :n/a
• favorite domain : to complete
• references :
– [Bleuzen-Guernalec and Colmerauer, 1997]: Narrowing a block of sortings in quadratic time
– [Mehlhorn and Thiel, 2000]: Faster algorithms for bound-consistency of the Sortedness and
the Alldifferent constraint
– global constraint catalog: count
Example:
CPModel m = new CPModel();
int n = 3;
IntegerVariable[] x = makeIntVarArray("x", n, 0, n);
IntegerVariable[] y = makeIntVarArray("y", n, 0, n);
m.addConstraint(sorting(x, y));
m.addConstraint(allDifferent(x));
CPSolver s = new CPSolver();
s.read(m);
s.solveAll();
8.103
startsAfter (constraint)
startsAfter(T, c) states that the task variable t starts after time c:
T.start ≥ c
• API :startsAfter(final TaskVariable t, final int c)
• return type : Constraint
• options : n/a
• favorite domain : n/a.
Examples: to complete
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8.104. STARTSAFTERBEGIN (CONSTRAINT)
8.104
startsAfterBegin (constraint)
startsAfterBegin(T1 , T2 , c) states that task T1 starts after the start time of T2 minus c:
T1 .start ≥ T2 .start − c
• API :startsAfterBegin(TaskVariable t1, TaskVariable t2, int c)
• return type : Constraint
• options : n/a
• favorite domain : n/a.
Examples: to complete
8.105
startsAfterEnd (constraint)
startsAfterEnd(T1 , T2 , c) states that task T1 starts after the end time of T2 minus c:
T1 .start ≥ T2 .end − c
• API :startsAfterEnd(TaskVariable t1, TaskVariable t2, int c)
• return type : Constraint
• options : n/a
• favorite domain : n/a.
Examples: to complete
8.106
startsBefore (constraint)
startsBefore(T, c) states that the task variable T starts before time c:
T.start ≤ c
• API :startsBefore(final TaskVariable t, final int c)
• return type : Constraint
• options : n/a
• favorite domain : n/a.
Examples: to complete
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CHAPTER 8. CONSTRAINTS (MODEL)
8.107
startsBeforeBegin (constraint)
startsBeforeBegin(T1 , T2 , c) states that task T1 starts before the start time of T2 minus c:
T1 .start ≤ T2 .start − c
• API :startsBeforeBegin(TaskVariable t1, TaskVariable t2, int c)
• return type : Constraint
• options : n/a
• favorite domain : n/a.
Examples: to complete
8.108
startsBeforeEnd (constraint)
startsBeforeend(T1 , T2 , c) states that task T1 starts before the end time of T2 minus c:
T1 .start ≤ T2 .end − c
• API :startsBeforeEnd(TaskVariable t1, TaskVariable t2, int c)
• return type : Constraint
• options : n/a
• favorite domain : n/a.
Examples: to complete
8.109
startsBetween (constraint)
startsBetween(T, c1 , c2 ) states that task T starts between times c1 and c2 :
c1 ≤ T.start ≤ c2
• API :startsBetween(TaskVariable t, int min, int max)
• return type : Constraint
• options : n/a
• favorite domain : n/a.
Examples: to complete
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8.110. STRETCHPATH (CONSTRAINT)
8.110
stretchPath (constraint)
A stretch in a sequence x is a maximum subsequence of (consecutive) identical values.
stretchPath(param, x) enforces the minimal and maximal length of the stretches in sequence x
of any values given in param: Consider the sequence x as a concatenation of stretches x1 .x2 . . . xk
with v i and li being respectively the value and the length of stretch xi ,
∀i ∈ {1, . . . , k}, ∀j,
param[j][0] = v i
=⇒
param[j][1] ≤ li ≤ param[j][2]
Useful for Rostering Problems. stretchPath is implemented by a regular constraint that performs GAC. The bounds on the stretch lengths are defined by param a list of triples of integers:
[value, min, max] specifying the minimal and maximal lengths of any stretch of the corresponding value.
This API requires a Java library on automaton available on http://www.brics.dk/automaton/. (It is
contained in the Choco jar file.)
• API : stretchPath(List<int[]> param, IntegerVariable... x)
• return type : Constraint
• options :n/a
• favorite domain : to complete
• references :
– [Pesant, 2004]: A regular language membership constraint
– global constraint catalog: stretch path
Example:
Model m = new CPModel();
int n = 7;
IntegerVariable[] vars = makeIntVarArray("v", n, 0, 2);
//define the stretches
ArrayList<int[]> lgt = new ArrayList<int[]>();
lgt.add(new int[]{0, 2, 2}); // stretches of value 0 are of length 2
lgt.add(new int[]{1, 2, 3}); // stretches of value 1 are of length 2 or 3
lgt.add(new int[]{2, 2, 2}); // stretches of value 2 are of length 2
m.addConstraint(stretchPath(lgt, vars));
Solver s = new CPSolver();
s.read(m);
s.solve();
8.111
times (constraint)
times(x1 , x2 , x3 ) states that the third argument is equal to the product of the two arguments:
x3 = x1 × x2 .
• API:
– times(IntegerVariable x1, IntegerVariable x2, IntegerVariable x3)
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CHAPTER 8. CONSTRAINTS (MODEL)
– times(int x1, IntegerVariable x2, IntegerVariable x3)
– times(IntegerVariable x1, int x2, IntegerVariable x3)
• return type : Constraint
• option : n/a
• favorite domain: bound
Example:
Model m = new CPModel();
IntegerVariable x = makeIntVar("x", 1, 2);
IntegerVariable y = makeIntVar("y", 3, 5);
IntegerVariable z = makeIntVar("z", 3, 10);
m.addConstraint(times(x, y, z));
Solver s = new CPSolver();
s.read(m);
s.solve();
8.112
tree (constraint)
Let G = (V, A) be a digraph on V = {1, . . . , n}. G can be modeled by a sequence of domain variables
x = (x1 , . . . , xn ) ∈ V n – the successors variables – whose respective domains are given by Di = {j ∈
V | (i, j) ∈ A}. Conversely, when instantiated, x defines a subgraph Gx = (V, Ax ) of G with Ax =
{(i, xi ) | i ∈ V } ⊆ A. Such a subgraph has one particularity: any connected component of Gx contains
either no loop – and then it contains a cycle – or exactly one loop xi = i and then it is a tree of root i
(literally, it is an anti-arborescence as there exists a path from each node to i and i has a loop).
tree(x, restrictions) is a vertex-disjoint graph partitioning constraint. It states that Gx is a forest
(its connected components are trees) that satisfies some conditions specified by restrictions. tree
deals with several kinds of graph restrictions on:
• the number of trees
• the number of proper trees (a tree is proper if it contains more than 2 nodes)
• the weigth of the partition: the sum of the weights of the edges
• incomparability: some nodes in pairs have to belong to distinct trees
• precedence: some nodes in pairs have to belong to the same tree in a given order
• conditional precedence: some nodes in pairs have to respect a given order if they belong to
the same tree
• the in-degree of the nodes
• the time windows on nodes (given travelling times on arcs)
Many applications require to partition a graph such that each component contains exactly one resource
node and several task nodes. A typical example is a routing problem where vehicle routes are paths (a
path is a special case of tree) starting from a depot and delivering goods to several clients. Another
example is a local network where each computer has to be connected to one shared printer. Last, one
can cite the problem of reconstructing plylogeny trees. The constraint tree can handle these kinds of
problems with many additional constraints on the structure of the partition.
• API : tree(TreeParametersObject param)
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8.112. TREE (CONSTRAINT)
• return type : Constraint
• options :n/a
• favorite domain : to complete
• references :
– [Beldiceanu et al., 2008]: Combining tree partitioning, precedence, and incomparability constraints
– global constraint catalog: proper forest (variant)
The tree constraint API requires a particular Model object, named TreeParametersObject. It can
be created with the following parameters:
parameter
type
description
n
int
number of nodes in the initial graph G
nT ree
IntegerVariable
number of trees in the resulting forest Gx
nP roper
IntegerVariable
number of proper trees in Gx
objective
IntegerVariable
(bounded) total cost of Gx
graphs
List<BitSet[]>
graphs encoded as successor lists,
graphs[0] the initial graph G,
graphs[1] a precedence graph,
graphs[2] a conditional precedence graph,
graphs[3] an incomparability graph
matrix
List<int[][]>
matrix[0] the indegree of each node, and
matrix[1] the starting time from each node
travel
int[][]
the travel time of each arc
Example:
import choco.kernel.model.variables.tree.TreeParametersObject;
Model m = new CPModel();
int nbNodes = 7;
//1- create the variables involved in the partitioning problem
IntegerVariable ntree = makeIntVar("ntree",1,5);
IntegerVariable nproper = makeIntVar("nproper",1,1);
IntegerVariable objective = makeIntVar("objective",1,100);
//2- create the different graphs modeling restrictions
List<BitSet[]> graphs = new ArrayList<BitSet[]>();
BitSet[] succ = new BitSet[nbNodes];
BitSet[] prec = new BitSet[nbNodes];
BitSet[] condPrecs = new BitSet[nbNodes];
BitSet[] inc = new BitSet[nbNodes];
for (int i = 0; i < nbNodes; i++) {
succ[i] = new BitSet(nbNodes);
prec[i] = new BitSet(nbNodes);
condPrecs[i] = new BitSet(nbNodes);
inc[i] = new BitSet(nbNodes);
}
// initial graph (encoded as successors variables)
succ[0].set(0,true); succ[0].set(2,true); succ[0].set(4,true);
succ[1].set(0,true); succ[1].set(1,true); succ[1].set(3,true);
succ[2].set(0,true); succ[2].set(1,true); succ[2].set(3,true); succ[2].set(4,true);
succ[3].set(2,true); succ[3].set(4,true); // successor of 3 is either 2 or 4
succ[4].set(2,true); succ[4].set(3,true);
succ[5].set(4,true); succ[5].set(5,true); succ[5].set(6,true);
succ[6].set(3,true); succ[6].set(4,true); succ[6].set(5,true);
// restriction on precedences
prec[0].set(4,true); // 0 has to precede 4
prec[4].set(3,true); prec[4].set(2,true);
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CHAPTER 8. CONSTRAINTS (MODEL)
prec[6].set(4,true);
// restriction on conditional precedences
condPrecs[5].set(1,true); // 5 has to precede 1 if they belong to the same tree
// restriction on incomparability:
inc[0].set(6,true); inc[6].set(0,true); // 0 and 6 have to belong to distinct trees
graphs.add(succ);
graphs.add(prec);
graphs.add(condPrecs);
graphs.add(inc);
//3- create the different matrix modeling restrictions
List<int[][]> matrix = new ArrayList<int[][]>();
// restriction on bounds on the indegree of each node
int[][] degree = new int[nbNodes][2];
for (int i = 0; i < nbNodes; i++) {
degree[i][0] = 0; degree[i][1] = 2; // 0 <= indegree[i] <= 2
}
matrix.add(degree);
// restriction on bounds on the starting time at each node
int[][] tw = new int[nbNodes][2];
for (int i = 0; i < nbNodes; i++) {
tw[i][0] = 0; tw[i][1] = 100; // 0 <= start[i] <= 100
}
tw[0][1] = 15; // 0 <= start[0] <= 15
tw[2][0] = 35; tw[2][1] = 40; // 35 <= start[2] <= 45
tw[6][1] = 5; // 0 <= start[6] <= 5
matrix.add(tw);
//4- matrix for the travel time between each pair of nodes
int[][] travel = new int[nbNodes][nbNodes];
for (int i = 0; i < nbNodes; i++) {
for (int j = 0; j < nbNodes; j++) travel[i][j] = 100000;
}
travel[0][0] = 0; travel[0][2] = 10; travel[0][4] = 20;
travel[1][0] = 20; travel[1][1] = 0; travel[1][3] = 20;
travel[2][0] = 10; travel[2][1] = 10; travel[2][3] = 5; travel[2][4] = 5;
travel[3][2] = 5; travel[3][4] = 2;
travel[4][2] = 5; travel[4][3] = 2;
travel[5][4] = 15; travel[5][5] = 0; travel[5][6] = 10;
travel[6][3] = 5; travel[6][4] = 20; travel[6][5] = 10;
//5- create the input structure and the tree constraint
TreeParametersObject parameters = new TreeParametersObject(nbNodes, ntree, nproper,
objective
, graphs, matrix, travel);
Constraint c = Choco.tree(parameters);
m.addConstraint(c);
Solver s = new CPSolver();
s.read(m);
//6- heuristic: choose successor variables as the only decision variables
s.setVarIntSelector(new StaticVarOrder(s, s.getVar(parameters.getSuccVars())));
s.solveAll();
8.113
TRUE (constraint)
T RU E always returns true.
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8.114. XNOR (CONSTRAINT)
8.114
xnor (constraint)
xnor(b1 , b2 ) states that two booleans are either both true, or both false:
(b1 = 1)
⇐⇒
(b2 = 1)
• API : xnor(IntegerVariable b1, IntegerVariable b2)
• return type : Constraint
• options : n/a
• favorite domain : n/a
Examples:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable v1 = makeIntVar("v1", 0, 1);
IntegerVariable v2 = makeIntVar("v2", 0, 1);
m.addConstraint(xnor(v1,v2));
s.read(m);
s.solve();
8.115
xor (constraint)
xor(b1 , b2 ) states that two booleans are the one true and the other one false:
(b1 = 1)
⇐⇒
(b2 = 0)
• API : xor(IntegerVariable b_1, IntegerVariable b_2)
• return type : Constraint
• options : n/a
• favorite domain : n/a
Examples:
Model m = new CPModel();
Solver s = new CPSolver();
IntegerVariable v1 = makeIntVar("v1", 0, 1);
IntegerVariable v2 = makeIntVar("v2", 0, 1);
m.addConstraint(xor(v1,v2));
s.read(m);
s.solve();
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Chapter 9
Options (Model)
This section lists and details the options that can be declared on variables or constraints within a Choco
Model.
9.1
Options and settings
The variables and some constraints allow the declaration of options. Options and settings are defined
in the classes Options and SettingType. Default options are specified. Most of the time, options
parameters are varargs.
9.1.1
Options for variables:
• Options.NO_OPTION, "", or empty argument
• Options.V_BOUND or "cp:bound"
– goal : force the solver to create bounded domain variable. It is a domain where only bound
propagation can be done (no holes). It is very well suited when constraints performing only
Bound Consistency are added on the corresponding variables. It must be used when large
domains are needed. Implemented by two integers.
– scope :
∗ IntegerVariable
∗ SetVariable
∗ TaskVariable
• Options.V_ENUM or "cp:enum"
– goal : force the solver to create enumerated domain variable (default option). It is a domain
in which holes can be created by the solver. It should be used when discrete and quite
small domains are needed and when constraints performing Arc Consistency are added on the
corresponding variables. Implemented by a BitSet object.
– scope :
∗ IntegerVariable
∗ SetVariable
∗ TaskVariable
• Options.V_BTREE or "cp:btree"
– goal : force the solver to create binary tree domain variable. Under development
– scope : IntegerVariable
• Options.V_BLIST or "cp:blist"
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CHAPTER 9. OPTIONS (MODEL)
– goal : force the solver to create bipartite list domain variable. It is a domain where unavailable
values are placed in the left part of the list, the other one on the right one.
– scope : IntegerVariable
• Options.V_LINK or "cp:link"
– goal : force the solver to create linked list domain variable. It is an enumerated domain where
holes can be done and every values has a link to the previous value and to the next value. It
is built by giving its name and its bounds: lower bound and upper bound. It must be used
when the very small domains are needed, because although linked list domain consumes more
memory than the BitSet implementation, it can provide good performance as iteration over
the domain is made in constant time. Implemented by a LinkedList object.
– scope : IntegerVariable
• Options.V_MAKESPAN or "cp:makespan"
– goal : declare the current variable as makespan
– scope : IntegerVariable
• Options.V_NO_DECISION or "cp:no decision"
– goal : force variable to be removed from the pool of decisional variables of the default search
strategy. Overriding the default search strategy will automatically erase this option.
– scope :
∗
∗
∗
∗
IntegerVariable
SetVariable
RealVariable
TaskVariable
• Options.V_OBJECTIVE or "cp:objective"
– goal : declare the objective variable
– scope :
∗ IntegerVariable
∗ SetVariable
∗ RealVariable
9.1.2
Options for expressions:
• Options.NO_OPTION, "", or empty argument
• Options.E_DECOMP or "cp:decomp"
– goal : force decomposition of the scoped expression.
– scope : IntegerExpressionVariable
9.1.3
Options and settings for constraints:
• Options.NO_OPTION, "", or empty argument
• Options.C_EXT_AC3 or "cp:ac3"
– goal : to get AC3 algorithm (searching from scratch for supports on all values)
– scope :
∗ feasPairAC(String, IntegerVariable, IntegerVariable, List)
∗ infeasPairAC(String, IntegerVariable, IntegerVariable, List)
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∗ infeasPairAC(String, IntegerVariable, IntegerVariable, boolean[][])
∗ relationPairAC(String, IntegerVariable, IntegerVariable, BinRelation)
• Options.C_EXT_AC32 or "cp:ac32"
– goal : to get AC3rm algorithm (maintaining the current support of each value in a non
backtrackable way)
– scope :
∗
∗
∗
∗
∗
∗
feasPairAC(String, IntegerVariable, IntegerVariable, List)
feasTupleAC(String, List, IntegerVariable[])
infeasPairAC(String, IntegerVariable, IntegerVariable, List)
infeasPairAC(String, IntegerVariable, IntegerVariable, boolean[][])
relationPairAC(String, IntegerVariable, IntegerVariable, BinRelation)
relationTupleAC(String, IntegerVariable[], LargeRelation)
• Options.C_EXT_AC322 or "cp:ac322"
– goal : to get AC3 with the used of BitSet to know if a support still exists
– scope :
∗
∗
∗
∗
feasPairAC(String, IntegerVariable, IntegerVariable, List)
infeasPairAC(String, IntegerVariable, IntegerVariable, List)
infeasPairAC(String, IntegerVariable, IntegerVariable, boolean[][])
relationPairAC(String, IntegerVariable, IntegerVariable, BinRelation)
• Options.C_EXT_AC2001 or "cp:ac2001"
– goal : to get AC2001 algorithm (maintaining the current support of each value)
– scope :
∗
∗
∗
∗
∗
∗
feasPairAC(String, IntegerVariable, IntegerVariable, List)
feasTupleAC(String, List, IntegerVariable[])
infeasPairAC(String, IntegerVariable, IntegerVariable, List)
infeasPairAC(String, IntegerVariable, IntegerVariable, boolean[][])
relationPairAC(String, IntegerVariable, IntegerVariable, BinRelation)
relationTupleAC(String, IntegerVariable[], LargeRelation)
• Options.C_EXT_AC2008 or "cp:ac2008"
– goal : to get AC2008 algorithm (maintained by STR)
– scope :
∗ feasTupleAC(String, List, IntegerVariable[])
∗ infeasTupleAC(String, List, IntegerVariable[])
∗ relationTupleAC(String, IntegerVariable[], LargeRelation)
• Options.C_EXT_FC or "cp:fc"
– goal : set filter policy to forward checking
– scope :
∗ feasTupleAC(String, List, IntegerVariable[])
∗ infeasTupleAC(String, List, IntegerVariable[])
∗ relationTupleAC(String, IntegerVariable[], LargeRelation)
• Options.C_ALLDIFFERENT_AC or "cp:ac"
– goal : for Regin implementation
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CHAPTER 9. OPTIONS (MODEL)
– scope : allDifferent(String, IntegerVariable[])
• Options.C_ALLDIFFERENT_BC or "cp:bc"
– goal : for bound all different using the propagator of A. Lopez-Ortiz, C.-G. Quimper, J.
Tromp, and P. van Beek. A fast and simple algorithm for bounds consistency of the alldifferent
constraint. IJCAI-2003
– scope : allDifferent(String, IntegerVariable[])
• Options.C_ALLDIFFERENT_CLIQUE or "cp:clique"
– goal : propagate on the clique of differences
– scope : allDifferent(String, IntegerVariable[])
• Options.C_GCC_AC or "cp:ac"
– goal : for Regin implementation
– scope : globalCardinality(String, IntegerVariable[], int[], int[], int)
• Options.C_GCC_BC or "cp:bc"
– goal : for Quimper implementation
– scope : globalCardinality(String, IntegerVariable[], int[], int[], int)
• Options.C_INCREASING_NVALUE_ATLEAST or "cp:atleast"
– goal : set filtering policy to filter on lower bound only
– scope : increasing nvalue(String, IntegerVariable, IntegerVariable[])
• Options.C_INCREASING_NVALUE_ATMOST or "cp:atmost"
– goal : set filtering policy to filter on upper bound only
– scope : increasing nvalue(String, IntegerVariable, IntegerVariable[])
• Options.C_INCREASING_NVALUE_BOTH or "cp:both"
– goal : set filtering policy to filter on lower and upper bound only
– scope : increasing nvalue(String, IntegerVariable, IntegerVariable[])
• Options.C_NTH_G or "cp:G"
– goal : global consistency
– scope :
∗ nth(String options, IntegerVariable index, int[] values, IntegerVariable val)
∗ nth(String option, IntegerVariable index, IntegerVariable[] varArray, IntegerVariable val)
∗ nth(String options, IntegerVariable index, IntegerVariable[] varArray, IntegerVariable val,
int offset)
• Options.C_CLAUSES_ENTAIL or "cp:entail"
– goal : ensures quick entailment tests
– scope : clause(IntegerVariable[],IntegerVariable[])
• Options.C_POST_PONED or "cp:postponed"
– goal : postponed a constraint
– scope : Constraint
• SettingType.ADDITIONAL_RULES or "cp:pack:ar"
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– goal : more filtering rules (recommended)
– scope :
∗ pack(int[] sizes, int nbBins, int capacity, String... options)
∗ pack(PackModeler modeler,String... options)
∗ pack(SetVariable[] itemSets, IntegerVariable[] loads, IntegerVariable[] bins, IntegerConstantVariable[] sizes, String... options)
∗ pack(SetVariable[] itemSets, IntegerVariable[] loads, IntegerVariable[] bins, IntegerConstantVariable[] sizes,IntegerVariable nbNonEmpty, String... options)
• SettingType.DYNAMIC_LB or "cp:pack:dlb"
– goal : feasibility test based on a dynamic lower bound
– scope :
∗ pack(int[] sizes, int nbBins, int capacity, String... options)
∗ pack(PackModeler modeler,String... options)
∗ pack(SetVariable[] itemSets, IntegerVariable[] loads, IntegerVariable[] bins, IntegerConstantVariable[] sizes, String... options)
∗ pack(SetVariable[] itemSets, IntegerVariable[] loads, IntegerVariable[] bins, IntegerConstantVariable[] sizes,IntegerVariable nbNonEmpty, String... options)
• SettingType.FILL_BIN or "cp:pack:fill"
– goal : dominance rule: fill a bin when an item fit into pertfectly equal-sized items and bins
must be equivalent
– scope :
∗ pack(int[] sizes, int nbBins, int capacity, String... options)
∗ pack(PackModeler modeler,String... options)
∗ pack(SetVariable[] itemSets, IntegerVariable[] loads, IntegerVariable[] bins, IntegerConstantVariable[] sizes, String... options)
∗ pack(SetVariable[] itemSets, IntegerVariable[] loads, IntegerVariable[] bins, IntegerConstantVariable[] sizes,IntegerVariable nbNonEmpty, String... options)
• SettingType.LAST_BINS_EMPTY or "cp:pack:lbe"
– goal : empty bins are the last ones
– scope :
∗ pack(int[] sizes, int nbBins, int capacity, String... options)
∗ pack(PackModeler modeler,String... options)
∗ pack(SetVariable[] itemSets, IntegerVariable[] loads, IntegerVariable[] bins, IntegerConstantVariable[] sizes, String... options)
∗ pack(SetVariable[] itemSets, IntegerVariable[] loads, IntegerVariable[] bins, IntegerConstantVariable[] sizes,IntegerVariable nbNonEmpty, String... options)
• SettingType.TASK_INTERVAL or "cp:cumul:ti"
– goal : for fast task intervals
– scope :
∗ cumulative(String name, TaskVariable[] tasks, IntegerVariable[] heights, IntegerVariable[]
usages, IntegerVariable consumption, IntegerVariable capacity, IntegerVariable uppBound,
String... options)
∗ cumulative(String name, TaskVariable[] tasks, IntegerVariable[] heights, IntegerVariable[]
usages, IntegerVariable consumption, IntegerVariable capacity, String... options)
∗ cumulative(String name, TaskVariable[] tasks, IntegerVariable[] heights, IntegerVariable
consumption, IntegerVariable capacity, String... options)
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CHAPTER 9. OPTIONS (MODEL)
• SettingType.SLOW_TASK_INTERVAL or "cp:cumul:sti"
– goal : for slow task intervals
– scope :
∗ cumulative(String name, TaskVariable[] tasks, IntegerVariable[] heights, IntegerVariable[]
usages, IntegerVariable consumption, IntegerVariable capacity, IntegerVariable uppBound,
String... options)
∗ cumulative(String name, TaskVariable[] tasks, IntegerVariable[] heights, IntegerVariable[]
usages, IntegerVariable consumption, IntegerVariable capacity, String... options)
∗ cumulative(String name, TaskVariable[] tasks, IntegerVariable[] heights, IntegerVariable
consumption, IntegerVariable capacity, String... options)
• SettingType.VILIM_CEF_ALGO or "cp:cumul:cef"
– goal : for Vilim theta lambda tree + lazy computation of the inner maximization of the edge
finding rule of Van hentenrick and Mercier
– scope :
∗ cumulative(String name, TaskVariable[] tasks, IntegerVariable[] heights, IntegerVariable[]
usages, IntegerVariable consumption, IntegerVariable capacity, IntegerVariable uppBound,
String... options)
∗ cumulative(String name, TaskVariable[] tasks, IntegerVariable[] heights, IntegerVariable[]
usages, IntegerVariable consumption, IntegerVariable capacity, String... options)
∗ cumulative(String name, TaskVariable[] tasks, IntegerVariable[] heights, IntegerVariable
consumption, IntegerVariable capacity, String... options)
• SettingType.VHM_CEF_ALGO_N2K or "cp:cumul:scef"
– goal : for Simple n2 ∗ k algorithm (lazy for R) (CalcEF – Van Hentenrick)
– scope :
∗ cumulative(String name, TaskVariable[] tasks, IntegerVariable[] heights, IntegerVariable[]
usages, IntegerVariable consumption, IntegerVariable capacity, IntegerVariable uppBound,
String... options)
∗ cumulative(String name, TaskVariable[] tasks, IntegerVariable[] heights, IntegerVariable[]
usages, IntegerVariable consumption, IntegerVariable capacity, String... options)
∗ cumulative(String name, TaskVariable[] tasks, IntegerVariable[] heights, IntegerVariable
consumption, IntegerVariable capacity, String... options)
• SettingType.OVERLOAD_CHECKING or "cp:unary:oc"
– goal : overload checking rule ( O(n*log(n)), Vilim), also known as task interval
– scope :
∗
∗
∗
∗
∗
disjunctive(TaskVariable[] tasks, String... options)
disjunctive(String name, TaskVariable[] tasks, String... options)
disjunctive(TaskVariable[] tasks,IntegerVariable[] usages, String... options)
disjunctive(String name, TaskVariable[] tasks,IntegerVariable[] usages, String... options)
disjunctive(String name, TaskVariable[] tasks,IntegerVariable[] usages, IntegerVariable
uppBound, String... options)
• SettingType.NF_NL or "cp:unary:nfnl"
– goal : NotFirst/NotLast rule ( O(n*log(n)), Vilim) (recommended).
– scope :
∗ disjunctive(TaskVariable[] tasks, String... options)
∗ disjunctive(String name, TaskVariable[] tasks, String... options)
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9.1. OPTIONS AND SETTINGS
∗ disjunctive(TaskVariable[] tasks,IntegerVariable[] usages, String... options)
∗ disjunctive(String name, TaskVariable[] tasks,IntegerVariable[] usages, String... options)
∗ disjunctive(String name, TaskVariable[] tasks,IntegerVariable[] usages, IntegerVariable
uppBound, String... options)
• SettingType.DETECTABLE_PRECEDENCE or "cp:unary:dp"
– goal : Detectable Precedence rule ( O(n*log(n)), Vilim).
– scope :
∗
∗
∗
∗
∗
disjunctive(TaskVariable[] tasks, String... options)
disjunctive(String name, TaskVariable[] tasks, String... options)
disjunctive(TaskVariable[] tasks,IntegerVariable[] usages, String... options)
disjunctive(String name, TaskVariable[] tasks,IntegerVariable[] usages, String... options)
disjunctive(String name, TaskVariable[] tasks,IntegerVariable[] usages, IntegerVariable
uppBound, String... options)
• SettingType.EDGE_FINDING_D or "cp:unary:ef"
– goal : disjunctive Edge Finding rule ( O(n*log(n)), Vilim) (recommended).
– scope :
∗
∗
∗
∗
∗
disjunctive(TaskVariable[] tasks, String... options)
disjunctive(String name, TaskVariable[] tasks, String... options)
disjunctive(TaskVariable[] tasks,IntegerVariable[] usages, String... options)
disjunctive(String name, TaskVariable[] tasks,IntegerVariable[] usages, String... options)
disjunctive(String name, TaskVariable[] tasks,IntegerVariable[] usages, IntegerVariable
uppBound, String... options)
• SettingType.DEFAULT_FILTERING or "cp:unary:df"
– goal : use filtering algorithm proposed by Vilim. nested loop, each rule is applied until it
reach it fixpoint.
– scope :
∗
∗
∗
∗
∗
disjunctive(TaskVariable[] tasks, String... options)
disjunctive(String name, TaskVariable[] tasks, String... options)
disjunctive(TaskVariable[] tasks,IntegerVariable[] usages, String... options)
disjunctive(String name, TaskVariable[] tasks,IntegerVariable[] usages, String... options)
disjunctive(String name, TaskVariable[] tasks,IntegerVariable[] usages, IntegerVariable
uppBound, String... options)
• SettingType.VILIM_FILTERING or "cp:unary:vf"
– goal : use filtering algorithm proposed by Vilim. nested loop, each rule is applied until it
reach it fixpoint.
– scope :
∗
∗
∗
∗
∗
disjunctive(TaskVariable[] tasks, String... options)
disjunctive(String name, TaskVariable[] tasks, String... options)
disjunctive(TaskVariable[] tasks,IntegerVariable[] usages, String... options)
disjunctive(String name, TaskVariable[] tasks,IntegerVariable[] usages, String... options)
disjunctive(String name, TaskVariable[] tasks,IntegerVariable[] usages, IntegerVariable
uppBound, String... options)
• SettingType.SINGLE_RULE_FILTERING or "cp:unary:srf"
– goal : use filtering algorithm proposed by Vilim. nested loop, each rule is applied until it
reach it fixpoint. A single filtering rule (debug only).
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– scope :
∗
∗
∗
∗
∗
9.1.4
disjunctive(TaskVariable[] tasks, String... options)
disjunctive(String name, TaskVariable[] tasks, String... options)
disjunctive(TaskVariable[] tasks,IntegerVariable[] usages, String... options)
disjunctive(String name, TaskVariable[] tasks,IntegerVariable[] usages, String... options)
disjunctive(String name, TaskVariable[] tasks,IntegerVariable[] usages, IntegerVariable
uppBound, String... options)
Options for solvers:
• Options.NO_OPTION, "", or empty argument
• Options.S_MULTIPLE_READINGS or "cp:multiple readings"
– goal : Allow a solver to read a model more than one time. In that case, the redundant
constraints for scheduling must be posted explicitly.
– scope : CPSolver
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Chapter 10
Branching strategies (Solver)
This section lists and details the branching strategies currently available in Choco.
10.1
AssignInterval (Branching strategy)
AssignInterval is a binary branching assigning two distinct intervals to a real variable. Following the interval bisection rule, the interval representing the domain of the selected variable is split
into two parts at its midpoint. In the first branch, the variable upper bound is set to the midpoint;
in the second branch, the variable lower bound is set to the midpoint.
B1 : x ∈ [x, m],
B2 : x ∈ [m, x],
with m =
x+x
2
• Constructor :AssignInterval(VarSelector<RealVar> varSel, ValIterator<RealVar> valIt)
• type of variable : real
• references : n/a
Example:
10.2
AssignOrForbidIntVarVal (Branching strategy)
AssignOrForbidIntVarVal is a binary branching assigning a value to an integer variable. In the
first branch, the selected value is assigned to the selected variable; in the second branch, the value
is removed from the variable domain.
B1 : x = v,
B2 : x 6= v
• Constructor :AssignOrForbidIntVarVal(VarSelector<IntDomainVar> varSel, ValSelector<IntDomainVar
> valSel)
• type of variable : integer
• references : n/a
Example:
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CHAPTER 10. BRANCHING STRATEGIES (SOLVER)
10.3
AssignOrForbidIntVarValPair (Branching strategy)
AssignOrForbidIntVarValPair is a binary branching assigning a value to an integer variable.
In the first branch, the selected value is assigned to the selected variable; in the second branch,
the value is removed from the variable domain. It requires a VarValPairSelector selecting both
variable and value at the same time.
B1 : x = v,
B2 : x 6= v
• Constructor :AssignOrForbidIntVarValPair(VarValPairSelector varValSel)
• type of variable : integer
• references : n/a
• see also : AssignOrForbidIntVarVal based on distinct variable/value selectors
Example:
10.4
AssignSetVar (Branching strategy)
AssignSetVar is a n-ary branching assigning distinct values to a set variable. The selected
variable is successively assigned, in each branch, to a next selected value.
B 1 : x = v1 ,
B2 : x = v2 ,
...,
B m : x = vm
• Constructor :AssignSetVar(VarSelector<SetVar> varSel, ValSelector<SetVar> valSel)
• type of variable : set
• references : n/a
Example:
10.5
AssignVar (Branching strategy)
AssignVar is an n-ary branching assigning distinct values to an integer variable. The selected
variable is successively assigned, in each branch, to a next selected value.
B 1 : x = v1 ,
B2 : x = v2 ,
...,
B m : x = vm
• Constructor :
– AssignVar(VarSelector<IntDomainVar> varSel, ValSelector<IntDomainVar> valSel)
– AssignVar(VarSelector<IntDomainVar> varSel, ValIterator<IntDomainVar> valIt)
• type of variable : integer
• references : n/a
Example:
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10.6. DOMOVERWDEGBRANCHINGNEW (BRANCHING STRATEGY)
10.6
DomOverWDegBranchingNew (Branching strategy)
DomOverWDegBranchingNew is a n-ary branching assigning distinct values to an integer variable.
It maintains (incrementally or dynamically) on each constraint, the count of the failures caused by
the constraint from the beginning of the search. To each variable are then associated, at any time,
three values: dom the current domain size, deg the current number of uninstantiated constraints
involving the variable, and w the sum of the counters associated with these deg constraints. The
strategy selects the variable with the smallest ratio ri = dom/w ∗ deg. Ties are randomly broken
when seed != null. The variable is then successively assigned, in each branch, to a next selected
value.
B 1 : x = v1 ,
B2 : x = v2 ,
...,
B m : x = vm ,
with x = arg min
domx wx ∗ degx
• Constructor :DomOverWDegBranchingNew(Solver s, IntDomainVar[] vars, ValIterator valIt, Number
seed)
• type of variable : integer
• references : [Boussemart et al., 2004]: Boosting systematic search by weighting constraints.
Example:
10.7
DomOverWDegBinBranchingNew (Branching strategy)
DomOverWDegBranchingNew is a binary branching assigning distinct values to an integer variable.
It maintains (incrementally or dynamically) on each constraint, the count of the failures caused by
the constraint from the beginning of the search. To each variable are then associated, at any time,
three values: dom the current domain size, deg the current number of uninstantiated constraints
involving the variable, and w the sum of the counters associated with these deg constraints. The
strategy selects the variable with the smallest ratio ri = dom/w ∗ deg. Ties are randomly broken
when seed != null. The variable is then assigned, in the first branch, to the selected value; in the
second branch, the value is removed from the variable domain.
B1 : x = v,
B2 : x 6= v,
with x = arg min
domx wx ∗ degx
• Constructor: DomOverWDegBinBranchingNew(Solver s, IntDomainVar[] vars, ValSelector valSel
, Number seed)
• type of variable : integer
• references : [Boussemart et al., 2004]: Boosting systematic search by weighting constraints.
Example:
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CHAPTER 10. BRANCHING STRATEGIES (SOLVER)
10.8
ImpactBasedBranching (Branching strategy)
ImpactBasedBranching is a n-ary branching assigning all distinct values to an integer variable.
The impact of a branching decision measures the reduction of the search space induced when the
decision was posted and propagated since the beginning of the search. Here, branching decisions are
variable-value assignments. The strategy selects an integer variable maximizing the total impact
minus the domain size. The variable is then successively assigned, in each branch, to its possible
values selected by decreasing order of their impacts. Restarting search can dramatically improve
performance.
B1 : x = v1 , B2 : x = v2 , . . . , Bm : x = vm ,
P
with x = arg max v∈D(x) (impact(x, v) − 1) and impact(x, vi ) > impact(x, vj ), ∀j < i.
• Constructor :
– ImpactBasedBranching(Solver s, IntDomainVar[] vars)
– ImpactBasedBranching(Solver s, IntDomainVar[] vars, AbstractImpactStrategy ibs)
• type of variable : integer
• references : [Refalo, 2004]: Impact-Based Search Strategies for Constraint Programming.
Example:
10.9
PackDynRemovals (Branching strategy)
PackDynRemovals is a n-ary branching for packing problems with identical bin capacities and
without pre-assignments, placing an item in distinct bins. It is a specialization of AssignVar for
breaking symmetries: the selected item x is successively placed, in each branch, in a next selected
bin. At every backtrack, once a bin has been tried, all other bins having exactly the same residual
space rcap become unselectable.
B1 : bin(x) = v1 ,
B2 : bin(x) = v2 ,
...,
Bn : bin(x) = vn ,
with rcap(vj ) 6= rcap(vi ) ∀i < j
• Constructor:PackDynRemovals(VarSelector varSel, ValSelector valSel, PackSConstraint pack)
• type of variable : integer (bin-to-item assignment variable)
• references : n/a
Example:
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10.10. SETTIMES (BRANCHING STRATEGY)
10.10
SetTimes (Branching strategy)
SetTimes is a n-ary branching for scheduling problems with makespan minimization objective,
fixing distinct task variables to their earliest starting time. At each node, a set of available tasks
is considered then, in each branch, one task variable is selected and its starting time is set to its
smallest value. The task variables are selected in order according to a TaskVarSelector, or in the
decreasing order defined by a Comparator, ties being randomly broken when boolean rand is set.
The search is not complete: do not use within a solveAll.
B1 : T1 .start = T1 .start,
B2 : T2 .start = T2 .start,
...,
Bn : Tn .start = Tn .start
• Constructor :
– SetTimes(Solver solver, List<TaskVar> tasks, TaskVarSelector varSel)
– SetTimes(Solver solver, List<TaskVar> tasks, Comparator<ITask> comp, boolean rand)
• type of variable : task
• references : n/a
Example:
10.11
TaskOverWDegBinBranching (Branching strategy)
TaskOverWDegBinBranching is a binary branching for scheduling problems, fixing the relative order between two task variables constrained by a precedence relation.
Like in
DomOverWDegBranchingNew, the pair of task variables is selected according to the ratio of the
size of the variable domains and the failure weight of the constraints involving these variables. In
the first branch, the selected pair of task variables is ordered according to the value returned by the
OrderingValSelector (1 if T1 precedes T2 or 0 if T2 precedes T1 ); in the second branch, the opposite
relative order is enforced.
B1 : precedence(T1 , T2 , order),
B2 : precedence(T1 , T2 , 1 − order)
• Constructor :TaskOverWDegBinBranching(Solver s, IPrecedenceRatio[] varRatios, OrderingValSelector
valOrd, Number seed)
• type of variable : task (actually: precedence boolean indicator)
• references : [Boussemart et al., 2004]: Boosting systematic search by weighting constraints.
Example:
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Chapter 11
Variable selectors (Solver)
This section lists and details the variable selectors currently available in Choco.
11.1
CompositeIntVarSelector (Variable selector)
CompositeIntVarSelector(h1 , h2 ) selects a constraint according to heuristic h1 , then selects an
integer variable involved in the constraint according to heuristic h2 :
h2 (support(h1 ))
• Constructor :CompositeIntVarSelector(ConstraintSelector h1, HeuristicIntVarSelector h2)
– ConstraintSelector is an interface. No implementation provided.
– HeuristicIntVarSelector is an abstract class implemented by: MinDomain, MaxDomain, MaxRegret, MaxValueDomain, MinValueDomain, MostConstrained. list to complete
• type of variable : integer
• references : n/a
Example:
11.2
CyclicRealVarSelector (Variable selector)
CyclicRealVarSelector(x) selects the real variables in the order they appear in array x, in a cyclic
way until they are all instantiated, i.e. with interval domain under the desired precision (static).
• Constructor :
– CyclicRealVarSelector(Solver s)
– CyclicRealVarSelector(Solver s, RealVar[] vars)
• type of variable : real
• references : n/a
Example:
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CHAPTER 11. VARIABLE SELECTORS (SOLVER)
11.3
LexIntVarSelector (Variable selector)
LexIntVarSelector(h1 , h2 ) selects the integer variable according to the first heuristic h1 , ties being
broken by the second heuristic h2 :
(
h1 (x)
if |h1 (x)| = 1,
h2 (h1 (x))
otherwise.
• Constructor: LexIntVarSelector(TiedIntVarSelector h1, HeuristicIntVarSelector h2)
– TiedIntVarSelector is an interface implemented by : MinDomain, MaxDomain, MaxRegret, MaxValueDomain, MinValueDomain, MostConstrained, RandomIntVarSelector.
– HeuristicIntVarSelector is an abstract class implemented by: MinDomain, MaxDomain, MaxRegret, MaxValueDomain, MinValueDomain, MostConstrained, RandomIntVarSelector.
• type of variable : integer
• references : n/a
Example:
11.4
MaxDomain (Variable selector)
MaxDomain(x) selects the integer variable with the largest domain (dynamic):
max |D(x)|
• Constructor :
– MaxDomain(Solver s)
– MaxDomain(Solver s, IntDomainVar[] vars)
• type of variable : integer
• references : n/a
Example:
11.5
MaxDomSet (Variable selector)
MaxDomSet(x) selects the set variable with the largest open domain (dynamic):
max |Env(x) \ Ker(x)|
• Constructor :
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11.6. MAXREGRET (VARIABLE SELECTOR)
– MaxDomSet(Solver s)
– MaxDomSet(Solver s, SetVar[] vars)
• type of variable : set
• references : n/a
Example:
11.6
MaxRegret (Variable selector)
MaxRegret(x) selects the integer variable with the largest difference between the two smallest values
in its domain (dynamic):
max(x2 − x),
with x2 = min(D(x) \ x)
• Constructor :
– MaxRegret(Solver s)
– MaxRegret(Solver s, IntDomainVar[] vars)
• type of variable : integer
• references : n/a
Example:
11.7
MaxRegretSet (Variable selector)
MaxRegretSet(x) selects the set variable with the largest difference between the two smallest values
in its envelope (dynamic):
max(x2 − x),
with x = min(Env(x)), x2 = min(Env(x) \ x)
• Constructor :
– MaxRegretSet(Solver s)
– MaxRegretSet(Solver s, SetVar[] vars)
• type of variable : set
• references : n/a
Example:
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CHAPTER 11. VARIABLE SELECTORS (SOLVER)
11.8
MaxValueDomain (Variable selector)
MaxValueDomain(x) selects the integer variable with the largest value in its domain (dynamic):
max(x̄)
• Constructor :
– MaxValueDomain(Solver s)
– MaxValueDomain(Solver s, IntDomainVar[] vars)
• type of variable : integer
• references : n/a
Example:
11.9
MaxValueDomSet (Variable selector)
MaxValueDomSet(x) selects the set variable with the largest value in its envelope (dynamic):
max(x̄),
with x̄ = max(Env(x))
• Constructor :
– MaxValueDomSet(Solver s)
– MaxValueDomSet(Solver s, SetVar[] vars)
• type of variable : set
• references : n/a
Example:
11.10
MinDomain (Variable selector)
MinDomain(x) selects the integer variable with the smallest domain (dynamic):
min |D(x)|
• Constructor :
– MinDomain(Solver s)
– MinDomain(Solver s, IntDomainVar[] vars)
• type of variable : integer
• references : n/a
Example:
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11.11. MINDOMSET (VARIABLE SELECTOR)
11.11
MinDomSet (Variable selector)
MinDomSet(x) selects the set variable with the smallest open domain (dynamic):
min |Env(x) \ Ker(x)|
• Constructor :
– MinDomSet(Solver s)
– MinDomSet(Solver s, SetVar[] vars)
• type of variable : set
• references : n/a
Example:
11.12
MinValueDomain (Variable selector)
MinValueDomain(x) selects the integer variable with the smallest value in its domain (dynamic):
min(x)
• Constructor :
– MinValueDomain(Solver s)
– MinValueDomain(Solver s, IntDomainVar[] vars)
• type of variable : integer
• references : n/a
Example:
11.13
MinValueDomSet (Variable selector)
MinValueDomSet(x) selects the set variable with the smallest value in its envelope (dynamic):
min(x),
with x = min(Env(x))
• Constructor :
– MinValueDomSet(Solver s)
– MinValueDomSet(Solver s, SetVar[] vars)
• type of variable : set
• references : n/a
Example:
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CHAPTER 11. VARIABLE SELECTORS (SOLVER)
11.14
MostConstrained (Variable selector)
MostConstrained(x) selects the integer variable involved in the largest number of constraints initially present in the solver (static):
max(initDeg(x))
• Constructor :
– MostConstrained(Solver s)
– MostConstrained(Solver s, IntDomainVar[] vars)
• type of variable : integer
• references : n/a
Example:
11.15
MostConstrainedSet (Variable selector)
MostConstrainedSet(x) selects the set variable involved in the largest number of constraints initially present in the solver (static):
max(initDeg(x))
• Constructor :
– MostConstrainedSet(Solver s)
– MostConstrainedSet(Solver s, SetVar[] vars)
• type of variable : set
• references : n/a
Example:
11.16
RandomIntVarSelector (Variable selector)
RandomIntVarSelector(x) selects an integer variable randomly (dynamic). The random seed can
be fixed.
• Constructor :
– RandomIntVarSelector(Solver s)
– RandomIntVarSelector(Solver s, long seed)
– RandomIntVarSelector(Solver s, IntDomainVar[] vars, long seed)
• type of variable : integer
• references : n/a
Example:
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11.17. RANDOMSETVARSELECTOR (VARIABLE SELECTOR)
11.17
RandomSetVarSelector (Variable selector)
RandomSetVarSelector(x) selects a set variable randomly (dynamic). The random seed can be
fixed.
• Constructor :
– RandomSetVarSelector(Solver s)
– RandomSetVarSelector(Solver s, long seed)
– RandomSetVarSelector(Solver s, SetVar[] vars, long seed)
• type of variable : set
• references : n/a
Example:
11.18
StaticSetVarOrder (Variable selector)
StaticSetVarOrder(x) selects the set variables in the order they appear in array x (static).
x1
• Constructor :
– StaticSetVarOrder(Solver s)
– StaticSetVarOrder(Solver s, SetVar[] vars)
• type of variable : set
• references : n/a
Example:
11.19
StaticVarOrder (Variable selector)
StaticVarOrder(x) selects the integer variables in the order they appear in array x (static).
x1
• Constructor :
– StaticVarOrder(Solver s)
– StaticVarOrder(Solver s, IntDomainVar[] vars)
• type of variable : integer
• references : n/a
Example:
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Chapter 12
Value iterators (Solver)
This section lists and details the value iterators currently available in Choco.
12.1
DecreasingDomain (Value iterator)
DecreasingDomain selects the integer variable largest value:
max(D(x))
• Constructor : DecreasingDomain()
• type of variable : integer
12.2
IncreasingDomain (Value iterator)
IncreasingDomain selects the integer variable smallest value:
min(D(x))
• Constructor : IncreasingDomain()
• type of variable : integer
12.3
RealIncreasingDomain (Value iterator)
RealIncreasingDomain selects the real variable smallest value:
min(D(x))
• Constructor : RealIncreasingDomain()
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CHAPTER 12. VALUE ITERATORS (SOLVER)
• type of variable : real
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Chapter 13
Value selector (Solver)
This section lists and details the value selectors currently available in Choco.
13.1
BestFit (Value selector)
BestFit, associated with a pack(items, load, bin, size) constraint, selects the bin v with the minimum residual space rcapv for the bin assignment integer variable bin[i] of item i:
X
size[j]
v ∈ D(bin[i]) : min{rcapv }
with rcapv = load[v] −
j∈Ker(items[v])
• Constructor : BestFit(PackSConstraint cstr)
• type of variable : integer (bin-to-item assignment variable bin[i])
13.2
CostRegularValSelector (Value selector)
CostRegularValSelector, associated with a CostRegular(z, hx1 , .., xn i, L(Π), hci,j i) constraint, selects a value vi in the domain of the sequence integer variable xi that can be extended to a solution
for the constraint hv1 , .., vn i ∈ L(Π) of maximum cost (if max, resp. minimum cost if ¬max):
vi ∈ D(xi ) :
∃vj ∈ D(xj ) ∀j 6= i, s.t.
n
X
cj,vj = z̄ (resp. = z)
j=1
Remember that the underlying data structure of a CostRegular constraint is a layered acyclic digraph,
where each arc in the i-th layer corresponds to a possible assignment of variable xi to a value v ∈ D(xi )
and has length civ , and where each path corresponds to a solution
Pn of the constraint, i.e. a word hv1 , .., vn i
of language L(Π), whose cost z is equal to the path length j=1 cj,vj .
This heuristic works with the ConstraintType.COSTREGULAR implementation of the constraint: see
FCostRegularValSelector for use with the ConstraintType.FASTCOSTREGULAR implementation (preferred).
• Constructor : CostRegularValSelector(CostRegular cr, boolean max)
• type of variable : integer
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CHAPTER 13. VALUE SELECTOR (SOLVER)
13.3
FCostRegularValSelector (Value selector)
FCostRegularValSelector, associated with a CostRegular(z, hx1 , .., xn i, L(Π), hci,j i) constraint,
selects a value vi in the domain of the sequence integer variable xi that can be extended to a solution
for the constraint hv1 , .., vn i ∈ L(Π) of maximum cost (if max, resp. minimum cost if ¬max):
vi ∈ D(xi ) :
∃vj ∈ D(xj ) ∀j 6= i, s.t.
n
X
cj,vj = z̄ (resp. = z)
j=1
Remember that the underlying data structure of a CostRegular constraint is a layered acyclic digraph,
where each arc in the i-th layer corresponds to a possible assignment of variable xi to a value v ∈ D(xi )
and has length civ , and where each path corresponds to a solution
Pn of the constraint, i.e. a word hv1 , .., vn i
of language L(Π), whose cost z is equal to the path length j=1 cj,vj .
This heuristic works with the ConstraintType.FASTCOSTREGULAR implementation of the constraint (preferred): see CostRegularValSelector for use with the ConstraintType.COSTREGULAR implementation.
• Constructor : FCostRegularValSelector(FastCostRegular cr, boolean max)
• type of variable : integer
13.4
MaxVal (Value selector)
MaxVal selects the largest value in the domain of the integer variable:
max(D(x))
• Constructor : MaxVal()
• type of variable : integer
13.5
MidVal (Value selector)
MidVal selects the closest value (equal or greater) to the integer variable domain midpoint:
min(v | v ≥
x+x
)
2
• Constructor : MidVal()
• type of variable : integer
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13.6. MINENV (VALUE SELECTOR)
13.6
MinEnv (Value selector)
MinEnv selects the smallest value in the open domain of the set variable:
min(Env(x) \ Ker(x))
• Constructor : MinEnv()
• type of variable : set
13.7
MinVal (Value selector)
MinVal selects the smallest value in the domain of the integer variable:
min(D(x))
• Constructor : MinVal()
• type of variable : integer
13.8
RandomIntValSelector (Value selector)
RandomIntValSelector selects a random value in the domain of the integer variable. The random
seed can be specified:
rand(D(x))
In case of bounded domain, it selects one of the bounds of the domain.
• Constructor :
– RandomIntValSelector()
– RandomIntValSelector(long seed)
• type of variable : set
13.9
RandomSetValSelector (Value selector)
RandomSetValSelector selects a random value in the open domain of the set variable. The random
seed can be specified:
rand(Env(x) \ Ker(x))
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CHAPTER 13. VALUE SELECTOR (SOLVER)
• Constructor :
– RandomSetValSelector()
– RandomSetValSelector(long seed)
• type of variable : set
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Chapter 14
Visualizer (Visualization)
14.1
Variable-oriented Visualizers
This section lists and details the variable-oriented visualizers currently available in Choco.
14.1.1
BinaryMatrix (visualizer)
BinaryMatrix(bool) is a specialized visualizer for a matrix of 0/1 variables bool.
• API :
– BinaryMatrix(IntDomainVar[][] bool, String display, int width, int height)
– BinaryMatrix(IntDomainVar[][] bool, String display, int x, int y, int width, int height
, String group, int min, int max)
14.1.2
BinaryVector (visualizer)
BinaryVector(bool) is a specialized visualizer for a vector of 0/1 variables bool.
• API :
– BinaryVector(IntDomainVar[] bool, String display, int width, int height)
– BinaryVector(IntDomainVar[] bool, String display, int x, int y, int width, int height,
String group, int min, int max)
14.1.3
DomainMatrix (visualizer)
DomainMatrix(xs) is a specialized visualizer for a matrix of variables xs.
• API :
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– DomainMatrix(Var[][] vars, String display, int width, int height)
– DomainMatrix(Var[][] vars, String display, int x, int y, int width, int height, String
group, int min, int max)
14.1.4
Vector (visualizer)
Vector(vars) is a specialized visualizer for variables vars.
• API :
– Vector(Var[] vars, String display, int width, int height)
– Vector(Var[] vars, String display, int x, int y, int width, int height, String group,
int min, int max)
14.1.5
VectorSize (visualizer)
VectorSize(vars) is specialized visualizer for the sum of cardinalities of variables vars.
• API :
– VectorSize(Var[] vars, String display, int width, int height)
– VectorSize(Var[] vars, String display, int x, int y, int width, int height, String group
, int min, int max)
14.1.6
VectorWaterfall (visualizer)
VectorWaterfall(vars is a specialized visualizer for a waterfall representation of a vector of variables vars.
• API :
– VectorWaterfall(Var[] vars, String display, int width, int height)
– VectorWaterfall(Var[] vars, String display, int x, int y, int width, int height, String
group, int min, int max)
14.2
Constraint-oriented Visualizers
This section lists and details the constraint-oriented visualizers currently available in Choco.
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14.2. CONSTRAINT-ORIENTED VISUALIZERS
14.2.1
AllDifferent (visualizer)
AllDifferent(vars) a specialized visualizer for the AllDifferent constraint, where vars is an array
of variables involved in one AllDifferent constraint.
• API :
– AllDifferent(IntDomainVar[] vars, String display, int width, int height)
– AllDifferent(IntDomainVar[] vars, String display, int x, int y, int width, int height,
String group, int min, int max)
14.2.2
AllDifferentMatrix (visualizer)
AllDifferentMatrix(vars) a specialized visualizer for a list of AllDifferent constraints, where
vars is a matrix of variables involved in n AllDifferent constraints.
• API :
– AllDifferentMatrix(IntDomainVar[][] vars, String display, int width, int height)
– AllDifferentMatrix(IntDomainVar[][] vars, String type, String display, int x, int y, int
width, int height, String group, int min, int max)
14.2.3
BooleanChanneling (visualizer)
BooleanChanneling(x, bs, o) is a specialized visualizer for the boolean channeling constraint, where
x is a variable, bs an array of boolean variables and o the first value (in Choco: domainChanneling).
• API :
– BoolChanneling(IntDomainVar x, IntDomainVar[] bs, int o, String display, int width, int
height)
– BoolChanneling(IntDomainVar x, IntDomainVar[] bs, String display, int width, int height
), where o is computed from the lower bound x’s domain.
– BoolChanneling(IntDomainVar var, IntDomainVar[] bool, int o, String display, int x, int
y, int width, int height, String group, int min, int max)
– BoolChanneling(IntDomainVar var, IntDomainVar[] bool, String display, int x, int y, int
width, int height, String group, int min, int max), where o is computed from the lower
bound x’s domain.
14.2.4
Cumulative (visualizer)
Cumulative(ts, l, e) is a specialized visualizer for the Cumulative constraint, where ts is an array of
task variables, l the limit variable and e the end variable.
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CHAPTER 14. VISUALIZER (VISUALIZATION)
• API :
– Cumulative(TaskVar[] tasks, IntDomainVar limit, IntDomainVar end, String display, int
width, int height), where display can take its value in {"compact", "expanded", "gantt"}
– Cumulative(TaskVar[] tasks, IntDomainVar limit, IntDomainVar end, String display, int
x, int y, int width, int height, String group, int min, int max), where display can take
its value in {"compact", "expanded", "gantt"}
14.2.5
Element (visualizer)
Element(index, values, value) is a specialized visualizer for the Element constraint, where index is
a variable, values an array of integer values and value a variable.
• API :
– Element(IntDomainVar index, int[] values, IntDomainVar value, String display, int width
, int height)
– Element(IntDomainVar index, int[] values, IntDomainVar value, String display, int x, int
y, int width, int height, String group, int min, int max)
14.2.6
Gcc (visualizer)
Gcc(vars, values, low, high) is a specialized visualizer for the Gcc constraint, where vars is an array
of variables, values an array of integer values, low minimum number of occurrences of values and
high maximum number of occurrences of values.
• API :
– Gcc(IntDomainVar[] vars, int[] values, int[] low, int[] high, String display, int width
, int height)
– Gcc(IntDomainVar[] vars, int[] values, int[] low, int[] high, String display, int x, int
y, int width, int height, String group, int min, int max)
14.3
Inverse (visualizer)
Inverse(X, Y ) is a specialized visualizer for the Inverse constraint, where X is an array of variables
and Y is an other array of variables.
• API :
– Inverse(IntDomainVar[] X, IntDomainVar[] Y, String display, int width, int height)
– Inverse(IntDomainVar[] X, IntDomainVar[] Y, String display, int x, int y, int width, int
height, String group, int min, int max)
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14.3. INVERSE (VISUALIZER)
14.3.1
LexLe (visualizer)
LexLe(X, Y ) is a specialized visualizer for the LexLe constraint, where X is an array of variables
and Y is an other array of variables.
• API :
– LexLe(IntDomainVar[] X, IntDomainVar[] Y, String display, int width, int height)
– LexLe(IntDomainVar[] X, IntDomainVar[] Y, String display, int x, int y, int width, int
height, String group, int min, int max)
14.3.2
LexLt (visualizer)
LexLt(X, Y ) is a specialized visualizer for the LexLt constraint, where X is an array of variables
and Y is an other array of variables.
• API :
– LexLt(IntDomainVar[] X, IntDomainVar[] Y, String display, int width, int height)
– LexLt(IntDomainVar[] X, IntDomainVar[] Y, String display, int x, int y, int width, int
height, String group, int min, int max)
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Part III
Extras
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Chapter 15
Sudoku and Constraint
Programming
15.1
Sudoku ?!?
Figure 15.1: A sudoku grid
Everybody knows those grids that appeared last year in the subway, in wating lounges, on colleague’s
desks, etc. In Japanese su means digit and doku, unique. But this game has been discovered by an
American ! The first grids appeared in the USA in 1979 (they were hand crafted). Wikipedia tells us
that they were designed by Howard Garns a retired architect. He died in 1989 well before the success
story of sudoku initiated by Wayne Gould, a retired judge from Hong-Kong. The rules are really simple:
a 81 cells square grid is divided in 9 smaller blocks of 9 cells (3 x 3). Some of the 81 are filled with one
digit. The aim of the puzzle is to fill in the other cells, using digits except 0, such as each digit appears
once and only once in each row, each column and each smaller block. The solution is unique.
15.1.1
Solving sudokus
Many computer techniques exist to quickly solve a sudoku puzzle. Mainly, they are based on backtracking
algorithms. The idea is the following: give a free cell a value and continue as long as choices remain
consistent. As soon as an inconsistency is detected, the computer program backtracks to its earliest past
choice et tries another value. If no more value is available, the program keeps backtracking until it can
go forward again. This systematic technique make it sure to solve a sudoku grid. However, no human
player plays this way: this needs too much memory !
see Wikipedia for a panel of solving techniques.
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CHAPTER 15. SUDOKU AND CONSTRAINT PROGRAMMING
15.2
Sudoku and Artificial Intelligence
Many techniques and rules have been designed and discovered to solve sudoku grids. Some are really
simple, some need to use some useful tools: pencil and eraser.
15.2.1
Simple rules: single candidate and single position
Figure 15.2: Simple rules: single candidates and single position
Let consider the grid on Figure 15.2 and the cell with the red dot. In the same line, we find: 3, 4, 6,
7, and 9. In the same column: 2, 3, 5, and 8. In the same block: 2, 7, 8, and 9. There remain only one
possibility: 1. This is the single candidate rule. This cell should be filled in with 1.
Now let consider a given digit: let’s say 4. In the block with a blue dot, there is no 4. Where can
it be ? The 4’s in the surrounding blocks heavily constrain the problem. There is a single position
possible: the blue dot. This another simple rule to apply.
Alternatively using these two rules allows a player to fill in many cells and even solve the simplest
grids. But, limits are easily reached. More subtle approaches are needed: but an important tool is now
needed ... an eraser !
15.2.2
Human reasoning principles
Figure 15.3: Introducing marks
Many techniques do exist but a vaste amount of them rely on simple principles. The first one is: do
not try to find the value of a cell but instead focus on values that will never be assigned to it. The
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15.2. SUDOKU AND ARTIFICIAL INTELLIGENCE
space of possibility is then reduced. This is where the eraser comes handy. Many players marks the
remaining possibilities as in the grid on the left.
Using this information, rather subtle reasoning is possible. For example, consider the seventh column
on the grid on the left. Two cells contain as possible values the two values 5 and 7. This means that
those two values cannot appear elsewhere in that very same column. Therefore, the other unassigned
cell on the column can only contain a 6. We have deduced something.
This was an easy to spot inference. This is not always the case. Consider the part
of the grid on the right. Let us consider the third column. For cells 4 and 5, only
two values are available: 4 and 8. Those values cannot be assigned to any other cell
in that column. Therefore, in cell 6 we have a 3, and thus and 7 in cell 2 and finally
a 1 in cell 3. This can be a very powerful rule.
Such a reasoning (sometimes called Naked Pairs) is easily generalized to any number
of cells (always in the same region: row, column or block) presenting this same
configuration. This local reasoning can be applied to any region of the grid. It is
important to notice that the inferred information can (and should) be used from a
region to another.
The following principles of human reasoning can be listed:
• reasoning on possible values for a cell (by erasing impossible ones)
• systematically applying an evolved local reasoning (such as the Naked Pairs rule)
• transmitting inferred information from a region to another related through a given a set of cells
15.2.3
Towards Constraint Programming
Those three principles are at the core of constraint programming a recent technique coming from
both artificial intelligence and operations research.
• The first principle is called domain reduction or filtering
• The second considers its region as a constraint (a relation to be verified by the solution of the
problem): here we consider an all different constraint (all the values must be different in a given
region). Constraints are considered locally for reasoning
• The third principle is called propagation: constraints (regions) communicate with one another
through the available values in variables (cells)
Constraint programming is able to solve this problem as a human would do. Moreover, a large
majority of the rules and techniques described on the Internet amount to a well-known problem: the
alldifferent problem. A constraint solver (as Choco) is therefore able to reason on this problem
allowing the solving of sudoku grid as a human would do although it has not be specifically designed to.
Ideally, iterating local reasoning will lead to a solution. However, for exceptionnaly hard grids, an
enumerating phase (all constraint solvers provide tools for that) relying on backtracking may be necessary.
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CHAPTER 15. SUDOKU AND CONSTRAINT PROGRAMMING
15.3
See also
• SudokuHelper a sudoku solver and helper applet developed with Choco.
• PalmSudoku a rather complete list of rules and tips for solving sudokus
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Glossary
Glossary
branching strategy heuristic controlling the execution of a search loop at a point where the control
flow may be split between different branches. 31
fix point, Fix point definition of a fix point. 43
search strategy composition of branching strategies. 31
Solver solver description. 27
value iterator heuristic specifying how to choose a value from a chosen variable, through an iterator,
at a fix point. 31
value selector heuristic specifying how to choose a value from a chosen variable at a fix point. 31
variable selector heuristic specifying how to choose a variable at a fix point. 31
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BIBLIOGRAPHY
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(2003). An efficient bounds consistency algorithm for the global-cardinality constraint. In International
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pages 600–614. Springer-Verlag.
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[Trick, 2003] Trick, M. A. (2003). A dynamic programming approach for consistency and propagation
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