Recent Advances on GPU Computing in Operations Research

Recent Advances on GPU Computing in Operations
Research
Vincent Boyer, Didier El Baz
To cite this version:
Vincent Boyer, Didier El Baz. Recent Advances on GPU Computing in Operations Research. IEEE
CPS. 2013 IEEE 27th International Symposium on Parallel & Distributed Processing Workshops
and PhD Forum, May 2013, BOSTON, United States. pp.1778-1787, <10.1109/IPDPSW.2013.45>.
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2013 IEEE 27th International Symposium on Parallel & Distributed Processing Workshops and PhD Forum
Recent Advances on GPU Computing in
Operations Research
Vincent Boyer 1,2 , Didier El Baz 2
Graduate Program in Systems Engineering
Universidad Autónoma de Nuevo León, Mexico
CNRS ; LAAS ; 7 avenue du colonel Roche, BP 54200 F-31031 Toulouse Cedex 4, France
Université de Toulouse, LAAS, F-31031 Toulouse France
Email: vincent.a.l.boyer@gmail.com, elbaz@laas.fr
1
2
The recent interest in GPU computing and hybrid computing (which is a combination of CPU and GPU computing), is
wide-spread. Almost all domains in science and engineering
are concerned. We can quote for example astrophysics,
seismic, oil industry and nuclear industry. Most of the time,
GPUs lead to dramatic improvements in the solution time
of practical problems.
It was quite natural for the Operations Research (OR)
community whose field of interest is prolific in difficult
problems to be attracted in GPU computing. In this paper, we
present recent advances on GPU computing in this domain.
Section II deals with some aspects related to GPU programming. Application of GPUs to OR is presented in the
sequel of the paper. In particular, contributions to Linear
Programming (LP) are presented in Section III. Section
IV deals with Integer Programming (IP). Conclusions and
Challenges are presented in Section V.
Abstract—In the last decade, Graphics Processing Units
(GPUs) have gained an increasing popularity as accelerators
for High Performance Computing (HPC) applications. Recent
GPUs are not only powerful graphics engines but also highly
threaded parallel computing processors that can achieve sustainable speedup as compared with CPUs. In this context,
researchers try to exploit the capability of this architecture
to solve difficult problems in many domains in science and
engineering. In this article, we present recent advances on GPU
Computing in Operations Research. We focus in particular on
Integer Programming and Linear Programming.
Keywords-GPU Computing; Operations Research; Integer
Programming; Linear Programming; Parallel Computing;
I. I NTRODUCTION
Originally designed for visualization purpose, graphics
accelerators, that are many cores parallel architectures, have
recently evolved towards powerful computing accelerators
in collaboration with CPU for High Performance Computing
(HPC) applications in science and engineering. In particular,
they have been widely applied to signal processing and linear
algebra.
We note that a device like the Tesla C2050 computing
processor with Fermi architecture has 448 computing cores
and 515 Gigaflops peak double precision floating point
performance [1]. As a consequence, many computer manufacturers like Dell, HP, SGI and Bull are currently using
Graphics Processing Units (GPUs) for acceleration purpose
in the clusters and systems they propose. Moreover, some
GPUs-based supercomputers like the Titan (17.6 Petaflops
with NVIDIA K20 GPUs) in the USA and Tianhe-1A (2.57
Petaflop/s with C2050 GPUs) in China are in the Top 10
supercomputers ranking. One can quote also the Nebulea
(1.27 Petalop/s with Intel X5650 processors and NVIDIA
Tesla C2050 GPUs) in China, the Tusbame 2.0 in Japan
and Roadrunner in the USA [2].
The exploitation of GPUs for HPC applications presents
many advantages:
• GPUs are powerful accelerators since they have now
hundreds of computing cores;
• GPUs are widely available and relatively chip;
• GPUs require less energy than other computing devices.
978-0-7695-4979-8/13 $26.00 © 2013 IEEE
DOI 10.1109/IPDPSW.2013.45
II. GPU C OMPUTING A ND H YBRID C OMPUTING
Thanks to high-level shading languages like DirectX or
OpenGL, graphics accelerators have started to be used for
non-graphical applications in the early 2000s. By that time,
problems like stock options pricing and protein folding
have been solved on graphics accelerators showing noticeable speedup. The acronym GPU was then introduced by
NVIDIA and people started to speak about General Purpose
computing on the GPU (GPGPU). Programming graphics
accelerators via graphics APIs turned out to be difficult since
basic programming features were missing and programs
were very complex (they had to be expressed in terms of
textures, graphics concepts and shader programs). We note
also that double precision floating point computation was
not possible in the beginning.
GPUs were reimagined as highly threaded streaming
processors when a new programming model extending C
with data-parallel constructs was proposed by Ian Buck
[3]. Concepts like kernels, streams and reduction operators
were then introduced. A new compiler and runtime system
permitted one to consider the GPU as a general-purpose
processor in a high-level language leading also to substancial
performance improvement.
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GPU
GeForce 7800 GTX
GeForce 8600 GTX
GeForce 9600 GT
GeForce GTX 260
GeForce GTX 280
GeForce GTX 285
GeForce GTX 295
GeForce GTX 480
Tesla C1060 [6]
T10 (Tesla S1070)
C2050 [1]
# cores
24
32
64
192
240
240
240
480
240
240
448
Clock (GHz)
0.58
0.54
0.65
1.4
1.296
1.476
1.24
1.4
1.3
1.44
1.15
architecture implements the IEEE 754-2008 floating-point
standard, providing the Fused Multiply-Add (FMA) instruction for both single and double precision arithmetic. Fermi
architecture has a two-level distributed thread scheduler. The
GigaThread global scheduler distributes thread blocks to SM
thread schedulers. Concurrent threads are created, managed
and run by the SM thread scheduler without overhead. This
permits one in particular to implement data parallelism.
NVIDIA has emphasized on the Single Instruction, Multiple
Threads (SIMT) parallel programming model. The reduction
in computation time resulting from exploiting GPUs for data
parallel applications can be dramatic. However, we note
that maximizing effective memory bandwidth of the GPU
and having good thread occupancy, i.e., giving sufficient
work to the SMs as well as ensuring coalesced memory
accesses for all cores of a given SM (in particular by
avoiding divergent branches) is particularly important in
order to obtain noticeable speedup with a GPU. It is also
important to note that thanks to CUDA or OpenCL, multiGPU computing is possible, i.e., the possibility to exploit
several accelerators in a unique application. This leads to
massive parallelism and the use GPUs for HPC applications.
Finally, we note that some libraries like the NVIDIA
CUDA Basic Linear Algebra Subroutines (cuBLAS) [8]
have been developped to help programmers to solve large
scale problems on the GPU. The cuBLAS library is a GPUaccelerated version of the complete standard BLAS library.
Memory (GB)
0.512
0.256
0.512
0.9
1
1
1
1.536
4
4
3
Table I
OVERVIEW OF NVIDIA GPU S QUOTED IN THE PAPER
The evolution of GPU’s hardware that permits one to
program more easily the device combined with the development in 2006 of Compute Unified Device Architecture,
CUDA (a software and hardware architecture that enables
the GPU to be programmed with some high level programming languages like C, C++ and Fortran) [4] or OpenCL
(a framework for writing programs that are executed across
heterogeneous platforms with CPUs, GPUs and other processors) [5] has fostered the popularity of GPU computing.
We recall that CUDA is a parallel computing platform and
programming model designed and developed by NVIDIA. It
permits one to increase computing performance by harnessing the power of the GPU. CUDA greatly simplifies GPU
programming. One merely writes a serial codes intended
to the CPU that calls parallel kernels defining the codes
to be implemented by threads on the GPU cores. CUDA
is based on a hierarchy of groups of threads and permits
one to use synchronization barrier. CUDA functionalities are
permanently extended in order to facilitate programming of
GPUs. Among the many advantages of CUDA one can quote
in particular: fast local memory that can be shared by a block
of threads, double precision floating point arithmetics and
more flexibility of coding than Graphics APIs. The recent
CUDA 5.0 [4] was designed to facilitate the dynamic use of
GPUs. Moreover, data transfers can now happen via highspeed network directly out of any GPU memory to any
other GPU memory in any other cluster without involving
assistance of the CPU.
Table I displays the characteristics of several GPUs
considered in the sequel. Up to very recently, the Fermi
architecture represented the last generation of NVIDIA GPU
architectures [7]. With 3.0 billion transistors, the Fermi
architecture features up to 512 CUDA cores; it is built
around a scalable array of multithreaded Streaming Multiprocessors (SM). The 512 CUDA cores are organized in 16
SMs. Typically, a Fermi multiprocessor consists of 32 Scalar
Processors cores. A CUDA core executes a floating point
or integer instruction per clock for a thread. Each CUDA
core or processor has a fully pipelined integer Arithmetic
Logic Unit (ALU) and Floating Point Unit (FPU). The Fermi
III. L INEAR P ROGRAMMING
In linear programming, the variables are nominally allowed to take a continuous range of values. The standard
form of such a problem can be formulated as follows:
min pT x,
s.t. Ax = b,
x ≥ 0,
(1)
(2)
(3)
where x is the vector of real numbers to be determined, A
is a matrix whose entries are fixed real contants, b and p are
two vectors of fixed real constants.
The simplex method has been designed by George
Dantzig [9] and different variants of the method have been
proposed in the literature for the solution of problem (1) (3). A complete review of the simplex method and its variants can be found in [10]. Basically, the simplex algorithm
starts from a feasible solution at a vertex of the polytope and
tries to improve the solution while preserving feasibility until
optimality is reached. In this section, we focus on several
approaches that have been inplemented on the GPU. Table
II summarizes the contributions in the literature.
A. The Simplex Tableau
In the simplex tableau algorithm, data are organized in a
tableau; pivoting operations are then applied until optimality.
1779
Algorithm
The Simplex Tableau
The Revised Simplex
The Interior Point Method
Year
2011
2011
2005
2009
2010
2008
and 27,000 constraints. The computational tests have shown
significant speedups. For example, a reduction of the computation time by a factor of 24.5 with two GPUs has been
observed for the largest instance. Meyer et al. [13] have used
a system with two Intel Xeon X5570 2.93GHz processor
and one NVIDIA Tesla S1070 computing system featuring
T10 GPUs. They have solved instances with up to 25,000
variables and 5,000 constraints and outperformed the opensource solver CLP [18] of the COIN-OR project.
Authors
Lalami et al. [11], [12]
Meyer et al. [13]
Greef [14]
Spampinato et al. [15]
Bieling et al. [16]
Jung and O’Leary [17]
Table II
L ITERATURE OVERVIEW IN L INEAR P ROGRAMMING AND GPU S
B. The Revised Simplex
This data structure is well suited to the GPU architecture: it
naturally tends to ensure efficient coalesced memory access
and does not need extra efforts for memory optimization.
In 2011, Lalami et al. have proposed a GPU implementation
of the simplex tableau algorithm via CUDA 2.3 in [11].
This implementation has been extended to the multi-GPU
context in [12]. The same year, Meyer et al. have proposed
a different multi-GPU implementation of the simplex tableau
algorithm in [13]. Both papers have dealt with a complete
implementation of the simplex algorithm on the GPUs
including the pivoting and the selection of the entering and
leaving variables in order to avoid extra communication
between the CPU and the GPUs. Multi-GPU computing
relies on problem decomposition. Several decomposition
schemes can be adopted. An horizontal decomposition distributes the constraints on the different GPUs. A vertical
decomposition distributes the variables of the LP problem on
the GPUs. Finally, one may consider also tiles. The choice of
a decompositon scheme has important consequences on the
resulting communication pattern and multi-GPU efficacy. A
decomposition based on tiles may appear scalable; it nevertheless necessitates many communications between GPUs.
In [13], the authors have adopted a vertical decomposition
in order to have less communication between GPUs. An
horizontal decomposition has been adopted in [12]. The
simplex tableau has been decomposed into parts that are
assigned to the different GPUs. An horizontal decomposition
of the simplex tableau presents the advantage to facilitate
the parallel processing of the entering variable column and
the ratio column (the leaving variable line is computed
by only one GPU at each iteration). The drawback of
this decomposition is to duplicate data. In [12], each GPU
updates only a part of the tableau. The work of each GPU
is managed by a distinct CPU thread. More precisely, CPU
threads are in charge of launching the kernels of the simplex
algorithm on each GPU, synchronising the work among
the different GPUs and sharing the results. This approach
presents the advantage to maintain the context of each CPU
thread all along the application, i.e., CPU threads are not
killed at the end of each simplex iteration. As a consequence,
communications tend to be minimized.
Lalami et al. [12] have used a server with Intel Xeon
E5640 2.66GHz processor and two NVIDIA C2050 GPUs.
They have considered instances with up to 27,000 variables
In the revised simplex method, only data that are relative
to the basic variables are stored in a tableau. This allows a
lower memory requirement than in the simplex tableau algorithm and suggests a reduction in complexity. All operations
of the simplex method can be seen as matrices operations.
Greeff [14] was the first to accelerate the revised simplex
method via GPU. He has achieved a speedup of 11.5 as
compared to an identical CPU implementation. Most of the
GPU computing drawbacks encountered by Greef in 2005
have been addressed since.
Later, Spampinato et al. [15] have tried to take benefit
from the advances in the linear algebra library cuBLAS [8].
They have used a system with Intel Core 2 Quad 2.83GHz
processor and NVIDIA GeForce GTX 280 GPU and have
reported a reduction in solution time by factor of 2.5 for
problems with 2,000 variables and 2,000 constraints when
compared to the ATLAS-based solver [19].
More recently, Bieling et al. [16] have proposed an
implementation of the revised simplex method which includes some algorithmic optimization, i.e., the steepest-edge
heuristic to select the entering variables [20] and an arbitrary
bound process in order to select the leaving variables. The
authors have reported a reduction in computation time by
a factor of 18 for instances with 8,000 variables and 2,700
constraints on a system with Intel Core 2 Duo E8400 3.0
GHz processor and NVIDIA GeForce 9600 GT GPU when
compared to results obtained with the GLPK solver [21].
C. The Interior Point Method
Interior point methods for solving problem (1) - (3) have
been considered for implementation on systems with GPUs.
We recall that these methods, also referred to as barrier
methods, reach the optimal solution of the LP by traversing
the interior of the feasible region.
Jung and O’Leary [17] have proposed a mixed precision
hybrid algorithm for solving LP using a primal-dual interior
point method. The algorithm is based on the rectangularpacked matrix storage scheme and uses the GPU for computationally intensive tasks like matrix assembly, Cholesky
factorization and forward and back substitution.
The hybrid algoritm was carried out on a system with
Intel Xeon 3.0GHz processor and NVIDIA GeForce 7800
GTX GPU. Instances with up to 4,000 variables and 1,000
1780
constraints have been considered; nevertheless it turns out
that the proposed hybrid algorithm has not clearly outperformed the sequential version on CPU due to data transfer
cost and communication latency.
IV. I NTEGER P ROGRAMMING
Integer Programming problems, IP, occur for example in
transportation, planning and logistics. In the standard form,
IP problems can be expressed as follows:
max pT x,
s.t. Ax = b,
x ≥ 0,
x integer.
(4)
(5)
(6)
(7)
where the entries of A, b and p are integer constants. Many
IP problems are NP-hard. Solution via dynamic programming and Branch and Bound is often considered in the
literature. The resulting data structure is often irregular; thus,
it is not well suited to GPU computing and solving IP with
the help of GPUs is in many case a challenge. We can find
two types of parallel approaches:
• either IP is entirely solved on GPU(s) through a specific
or adapted parallel algorithm;
• or GPUs are used to accelerate only the most time consuming activities or parts of codes (hybrid algorithms).
To the best of our knowledge, four types of problems
have been studied in the GPU literature: Knapsack Problems
(KP), Scheduling Problems (SP), Assignment Problems (AP)
and Travelling Salesman Problems (TSP). The proposed
approaches and computational results are presented in the
sequel. The literature is summarized in Table III.
A. Knapsack Problems
1) Dynamic programming: The solution of KP via a
hybrid dense dynamic programming algorithm implemented
with CUDA 2.0 has been considered in [22]. At each
step, computations in the loop that processes the classical
Bellman’s dynamic programming recursion (which is time
consuming) have been implemented in parallel on the device.
A data compression technique has also been proposed
in order to deal with the high memory requirement of the
dynamic programming method. This technique has permitted
the authors to reduce the memory occupancy needed to
reconstruct the optimal solution and the amount of data
transferred between the host and device.
Computational experiments have been carried out on a
system with Intel Xeon 3.0 GHz and NVIDIA GTX 260
GPU. Randomly generated correlated problems with up to
100,000 variables have been considered. We note that dense
dynamic programmic is known to be suited to correlated
instances. Computational results have shown that these
problems can be solved within relatively small computing
time via GPU (only few hundred seconds) and memory
occupancy. Moreover, a reduction in computation time by a
factor of 26 has been observed for instances with more than
40,000 variables. We note that the reduction in matrix size
is better when the size of the problem is increased, resulting
in a more efficient compression while the overhead does not
exceed 3% of the overall processing time.
The contribution in [22] has been further extended in [23],
where a multi-GPU hybrid implementation via CUDA 2.3 of
the dense dynamic programming method has been proposed.
The approach is well suited to the case where a CPU is
connected to several GPUs. The solution presented in [23] is
based on multithreading and the concurrent implementation
of kernels on GPUs; each kernel being associated with a
given GPU and managed by a CPU thread; the context of
each host thread being maintained all allong the application,
i.e., host threads are not killed at the end of each dynamic
programming step. This technique tends also to reduce data
exchanges between host and device. A load balancing procedure has been implemented in order to maintain efficiency
of the parallel algorithm.
Computational experiments have been carried out on
a machine with Intel Xeon 3 GHz processor, 1 GB
memory and Tesla S1070 computing system. Strongly
correlated problems with up to 100,000 variables have been
considered. Preliminary results have shown a reduction in
computation time by a factor of 14 with one GPU and 28
with two GPUs (without data compression techniques).
2) Branch and Bound: A hybrid Branch and Bound
algorithm has been proposed in [25] for the solution of the
knapsack problem. The GPU is used only when the number
of Branch and Bound nodes is important. Computation is
performed on the CPU when the number of nodes is under
a certain threshold. Elimination of nonpromising nodes and
concatenation of the list of nodes is always performed on
the CPU.
Dimension and capacity of the problem are stored in the
constant memory of the GPU since they do not change
during the solution of the problem. Weights and profits
of items are stored in the texture memory of the GPU
that is larger than the constant memory; both memories
presenting low latency. When in use, the GPU takes care
of the separation phase, i.e., creation of new nodes in
parallel, it also computes bounds in parallel and the best
lower bound via atomicMax operation. Finally, the GPU
performs in parallel bounds comparison and nonpromising
nodes labelling (the later task concerns nodes with an upper
bound smaller than the best lower bound).
Computational tests have been carried out on a system
with Intel Xeon E5640 2.66GHz processor and NVIDIA
C2050 GPU. Reduction in computation time by a factor
of 20 has been observed in [25] for strongly correlated
problems with 500 variables. The reduction in computation
time has been further improved by a factor of 52 in [26],
1781
Problem
KP
Algorithm
Dynamic Programming
Branch and Bound
Year
Authors
2011-2012 Boyer et al. [22], [23]
2012
Boukedjar et al. [24]
2012
Lalami et El Baz [25]
2012
Lalami [26]
Genetic Algorithm
2012
Pedemonte et al. [27]
SP
Tabu Search
2008
Janiak et al. [28]
2011
Czapińsky et al. [29]
2011
Luong et al. [30]
2012
Bukata [31]
2013
Bukata and S̆ucha [32]
Branch-and-Bound
2012
Chakroum et al. [33], [34]
2012
Melab et al. [35]
Genetic Algorithm
2011
Zajı́c̆ek and S̆ucha [36]
2011
Nesmachnow et al. [37]
2013
Pinel et al. [38]
AP
Tabu Search
2010
Luong et al. [39]
2011
Luong [40]
Genetic Algorithm
2009
Tsutsui et Fujimoto [41]
2010
Soca et al. [42]
2011
Tsutsui et Fujimoto [43]
Deep Greedy Switching 2011
Roverso et al. [44]
TSP
Ant Colonies
2007
Catala et al. [45]
2009
Li et al. [46]
2009
You [47]
2011
Cecilia et al. [48]
Max-Min Ant System
2009
Jiening et al. [49]
2009
Bai et al. [50]
2010
Fu et al. [51]
Genetic Algorithm
2011
Chen et al. [52]
Immune Algorithm
2009
Li et al. [53]
Tabu Search
2008
Janiak et al. [28]
Table III
L ITERATURE OVERVIEW IN I NTEGER P ROGRAMMING AND GPU C OMPUTING
for strongly correlated problems with 1,000 variables and
some optimization in access to the GPU memory. The
computational results have shown that the more difficult a
problem is, the larger the number of Branch and Bound
nodes and the more remarkable the reduction in time due
to GPU accelerator. The reader is also referred to [24] for
another contribution to this field.
that the reduction in computation time ranges from 5.09 to
35.7 times according to the size of the considered instances.
B. Scheduling Problems
1) Branch and Bound: The solution of the Flow-shop
Scheduling Problem (FSP) via parallel Branch and Bound
methods using GPU has been studied by several authors.
In [33], a selection operator based on the best-first strategy
is used until the work pool reaches a given size. Then, a
selection operator based on the depth-first strategy is used.
This technique permits one to provide enough work to the
GPU. The evaluation of bounds that is time consuming was
performed in parallel in [33]; an original technique was also
proposed to avoid divergent threads in a warp resulting from
conditional branches.
Computational experiments have been carried out via
CUDA 4.0 on a system with Intel Xeon E5520 2.27 GHz
bi-processor and NVIDIA C2050 computing system. Some
instances of flow-shop problems proposed by Taillard (see
[54]) that range from twenty jobs on twenty machines to
two hundred jobs on twenty machines have been considered.
Maximum speedup factor of 77 has been observed for
instances with two hundred jobs on twenty machines as
compared with a sequential version.
The parallel application of branching, bounding, selection
and elimination operators has been considered in [34] as well
3) Genetic Algorithm: Pedemente, Alba and Luna have
proposed in [27] a genetic algorithm specially designed to
run on GPU. The algorithm called Systolic Genetic Search
(SGS) is based on the model of systolic computation, i.e.,
the synchronous circulation of solutions through a grid
of processing units. At each iteration, the crossover and
mutation operators, the fitness function evaluation and the
elitist replacement are carried out on the GPU. The exchange
of directions operator can additionally be applied on the
GPU. The five kernels quoted above are invoked by the host.
Experiments have been carried out on a system with
Pentium D 3.0 GHz processor, 2 GB RAM and NVIDIA
GeForce GTX 480 GPU. Problems without correlation and
up to 1,000 variables have been considered. Experimental
results have shown that the SGS method produces solutions
of very good quality. GPU and CPU versions of SGS have
been carried out with the same seeds so that experimental
results were exactly the same. Numerical results have shown
1782
as exploiting higher parallelism via workload distribution on
a multi-GPU testbed. Workloads have been equally splitted
into as many groups as there are GPUs in the system and
an equal number of CPU threads has been created.
In [34], computational experiments have been carried out
via CUDA 4.0 on a system with Intel Xeon E5520 2.27 GHz
bi-processor and NVIDIA Tesla S1070 computing system.
The considered problems range from twenty jobs on ten
machines to two hundred jobs on twenty machines. Instances
have been solved on one GPU 11 to 78 times faster than with
a single CPU core. Morover, a maximum speedup factor of
105 has been observed with two GPUs.
Reference is also made to [35] for a study on a parallel
Branch and Bound method using GPU whereby the evaluation of lower bounds is made on the device while generation
of subproblems, i.e., elimination, selection and branching
operations is implemented on the host.
2) Genetic algorithms: Zajı́c̆ek and S̆ucha have studied
a parallel island-based Genetic Algorithm (GA) for the
solution of the FSP in [36]. According to the proposed
homogeneous model, all computations were carried out on
the GPU in order to reduce communication between CPU
and GPU. The authors have implemented a GA whereby
islands are essentially used for migration of individuals,
i.e., specific solutions among subsets of solutions, the socalled populations. Evaluations, mutations and crossovers of
solutions in the same subset of solutions were performed in
parallel and independently of other populations.
Experiments were performed on a system with AMD
Phenom II X4 945 3.0 GHz processor and NVIDIA Tesla
C1060 GPU. Some instances with one hundred activities and
five machines were solved 110 faster than with the AMD
CPU.
Nesmachnow and Canabé have considered the solution
of the Heterogenous Computing Scheduling Problem. A
parallel implementation of the Min-Min heuristic on GPU,
whereby the evaluation of the criteria for all machines is
made in parallel on the GPU for each unassigned task, was
proposed in [37] (a parallel version of the Sufferage heuristic
was also studied in the paper).
Computational experiments were carried out on a system
with Dell Xeon E5530 2.4 GHz processor and NVIDIA
C1060 GPU. Reduction of computational time by a factor
of 5 has been obtained for parallel Min-Min (5.5 for parallel
Sufferage).
Pinel et al. have considered the parallel solution of
scheduling of independant tasks problems in [38]. They have
also proposed implementations on GPU of the Min-Min
heuristic and GraphCell a parallel cellular genetic algorithm
(two new parallel recombination operators are also proposed
in the paper).
Computational experiments have been carried out on a
system with Intel Xeon E5440 2.83 GHz processor and
NVIDIA Tesla C2050 GPU. We note that significant speedup
has been obtained for the GPU version of the Min-Min
heuristic.
3) Tabu Search: Czapiński and Barnes have implemented
a Tabu Search (TS) metaheuristic method for the solution of
FSP via GPU in [29].
The TS method was carried out on a system with Intel
Xeon 3.0 GHz processor with 2GB memory and NVIDIA
Tesla C1060 GPU. The authors have claimed that their
implementation is 89 times faster than the CPU version.
In [30], Luong, Melab and Talbi have studied the implementation on GPU of an aggregated TS method for the FSP.
Their paper deals more generaly with the implementation
on GPU of multiobjective local search algorithms. The
generation of neighborhood was done on GPU in order
to reduce data transfers; several representations have been
considered (see also [40] and [55]).
Experiments have been carried out on two systems:
•
•
a system with Xeon 3.0 GHz processor and GTX 285
GPU;
a system with Core i7 3.2 GHz processor and GTX 480
GPU.
The considered problems range from twenty jobs and
ten machines to two hundred jobs and twenty machines.
The observed maximum speedup was 10 times for the first
system and 16 times for the second system.
We note that Pareto local search algorithms have also been
studied in [30]. The same instances have been considered
and the observed maximum speedup was 9.4 times for the
first system and 15.7 times for the second system.
Bukata and S̆ucha have subsequently presented a parallel
TS method for the Resource Constrained Project Scheduling
Problem (RCPSP) according to the proposed homogeneous
model whereby all computations are performed on the GPU
(see [32]). The Simple Tabu List implementation (with
constant algorithmic complexity) was specially designed for
the GPU, a new parallel algorithm for schedule evaluation
was proposed and parallel reductions were applied.
Experiments have been carried out via CUDA 3.2 on
a server with Intel Xeon E5640 2.66 GHz processor, 12
GB memory and NVIDIA Tesla C2050 GPU. Solution
interchange was performed via the global memory where
the global best solution and working set were stored. Simple
Tabu lists were also stored in the global memory. Local
memory was used for resource arrays and activities start time
values. J120 instances with 600 projects and 120 activities
have been used for performance comparison. Experimental
results have shown that the GPU is able to perform the
same number of iterations 55 times faster than the CPU in
average (see also [31]). Experiments have also shown that
the parallel TS method outperforms the CPU version on what
concerns the quality of solutions that is comparable to the
one obtained with efficient metaheuristics in the literature
(see [32]).
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C. Assignment Problems
instances of the TSPLIB library [57] are generally used as
benchmark instances for computational tests.
In this section, we focus on the ant colony approaches
which have been given a particular attention in the literature
for the solution of the TSP via GPUs. Ant colonies [58]
are population-based metaheuristics for solving optimization
problems. They use artificial ants to construct solutions by
considering pheromone trails that reflect the search procedure.
The implementation on GPU of some other metaheuristics
has been investigated for the TSP. The reader is referred
to the work of Janiak et al. on Tabu Search [28], Li et
al. on Immune Algorithm [53] and Chen et al. on Genetic
Algorithm [52]. To the best of our knowledge, no exact
approach has been addressed in the literature.
1) Ant Colonies Optimization (ACO): The first proposed
work in this area was due to Catala et al. in 2007 (see
[45]); the solution of the orienteering problem, also known
as the selective travelling salesperson problem (see [59]),
was considered. The results have shown that the proposed
approach that was implemented on a single GeForce 6600
GT GPU stood competitive with a parallel ACO running on
a GRID with up to 32 nodes.
In 2009, Li et al. [46] and You et al. [47] have proposed
GPU implementations of the ACO method with adequate
memory management.
The implementation of Li et al. is based on a fine-grain
model whereby ants are assigned to one thread.
You et al. have focused on the construction phase of
the ACO whereby each GPU thread builds a route for one
ant. Computational experiments have been carried out on
a system with Intel Core 2 Duo 2.20GHz processor and
GeForce 8600GT GPU. Problems with up to 800 cities have
been considered and a maximum speedup of 20 has been
observed.
In 2011, Cecilia et al. [48] have presented new results on
the parallelization of the ACO method on GPU. Different
strategies for the implementation of both stages of the ACO
algorithm on the GPU, i.e., the construction phase and the
pheromone update are discussed in the paper. The authors
have proposed to assign one ant to each block of threads
Moreover, each thread represents a city or a set of cities
the ant may visit. This strategy is followed in order to
overcome some drawbacks encountered by former methods
in the literature. For the pheromone update, the authors have
proposed a scatter-to-gather transformation (see [60]) which
avoids the use of atomic operations on GPU.
Computational experiments have been carried out on a
C1060 GPU. Speedup factors of 25 have been reported for
some instances with 2,396 cities.
2) Max-Min Ant System: In 2009, Jiening et al. have
proposed an implementation of a variant of the ACO method:
the Max-Min Ant System (MMAS) [49]. In this implementation, the tour construction stage is carried out on GPU,
1) Deep Greedy Switching: Roverso et al. have proposed
a GPU implementation of the Deep Greedy Switching
(DGS) heuristic for the solution of the Linear Sum Assignment Problem (LSAP) in [44]. Classically, agents have to
be assigned to an equal number of jobs while maximizing
the total benefit. Basically, the DGS heuristic starts from
a random initial solution and moves to better solutions by
considering a neighborhood resulting from a restricted 2exchange. Each agent tries to find the best solution from
a given neighborhood. The improvement, or difference, in
the objective function between the current and new solution, called agent difference evaluations and job difference
evaluations, that are computationally expensives have been
carried out in parallel on the GPU.
Computational experiments have been carried out on a
system with Core 2 Duo 2.4 GHz processor, 4 GB memory
and NVIDIA GTX 295 GPU. Randomly generated instances
with up to 9744 jobs have been considered. Reduction of
computation time by a factor of up to 27 has been observed.
2) Tabu Search: We note also that Luong et al. have
proposed a hybrid Tabu Search method for the 3-dimensional
Quadratic Assignment Problem (QAP) in [39]. The basic TS
algorithm runs on the CPU. Evaluation and neighborhood
generation that are time consumming run on the GPU.
Computational tests were performed on a system with
Intel Core Duo 3.0 GHz processor and NVIDIA 8600 GT
GPU. A reduction of computation time by a factor of 4 was
obtained.
3) Genetic Algorithm: Tsutsui and Fujimoto have proposed to solve the QAP via a parallel Ant Colony Optimization method (ACO) implemented on GPU in [43]. They
have considered 2-opt local search whereby moves can be
divided into two groups that can be computed in parallel by
blocks of threads. Pheromone update and sampling are also
carried out on the device.
Computational tests have been carried out on a system
with Intel Core i7 965 3.2 GHz processor and NVIDIA
GeForce GTX 480 GPU. Real life like instances and randomly generated instances of the QAPLIB library whose size
ranges from 50 to 150 have been considered. Speedups of
24.6 times have been observed as compared with a sequential
version of the method implemented on the CPU (reference
is also made to [41]).
Soca et al. [42] have proposed a framework for automatic
implementation of parallel cellular genetic algorithms on
GPU.
D. Travelling Salesman Problems
Giving a set of cities, the TSP [56] involves finding the
shortest route that visits each city exactly once. It is a
well-known NP-hard optimization problem and is used as
a standard benchmark for many heuristic algorithms. The
1784
presented important contributions to Integer Programming
and Linear Programming. We note that many OR problems
have been considered in the literature and that significant
speedups have been obtained for several exact methods or
heuristic algorithms thanks to the GPUs. At this point, it is
important to note that one cannot establish a quantitative
comparison between the proposed approaches since they
have led to implementations on several generations of GPUs.
For many applications including OR problems, the future
of GPU computing seems very promising. The new NVIDIA
Kepler architecture features teraflops of integer, single precision, and double precision performance and high memory
bandwidth [62]. Moreover, programming tools like CUDA
and OpenCL (or the recent OpenACC [63]) always tend to
facilitate programming and improve efficiency of this type
of architecture by hiding programming difficulties. We note
in particular that OpenACC is a set of high-level pragmas
that enables C/C++ and Fortran programmers to exploit
highly parallel processors with much of the convenience of
OpenMP. In this case, pragma are code annotations which
inform the compiler of structured loop or succeeding block
of code as a good candidate for parallelization.
We believe that OR industrial codes in the future will be
able take a great benefit from GPUs and to propose very
attractive and fast solutions to their users. An important
challenge remains in the exact solution of industrial
problems of significant size via GPUs.
and the shortest path is computed on the CPU.
Computational experiments have been carried out on a
system with AMD 2.79Hz processor and NVIDIA Quadro
Fx 4500 GPU. On a typical test instance with thirty cities,
a reduction of computation time by a factor of 1.4 was
observed.
In 2009, Bai et al. have discussed the implementation of
all phases of MMAS on GPU [50]. For the tour construction
stage each ant colony is assigned to a thread block, whilst for
the pheromone update, each city corresponds to one thread.
Computational experiments have been carried out on a
system with AMD Athlon Dual Core Processor 3600+ and
NVIDIA GeForce 8800 GTX GPU. Processing time twice
as fast as with CPU have been reported for instances with
400 cities.
More recently, Fu et al. have proposed in [51] an implementation of MMAS on GPU that makes use of the
Jacket toolbox which connects MATLAB to GPU (see [61]).
Ants share only one pseudorandom number matrix, one
pheromone matrix, one taboo matrix and one probability
matrix in order to reduce communication between CPU and
GPU. Furthermore, a variation of the traditional roulette
wheel selection (the All-In-Roulette which is well suited to
the GPU architecture) has been used.
Computational tests have been carried out on a system
with Intel i7 3.3GHz processor and Tesla C1060 GPU.
Speedups of 400 on GPU as compared with the sequential
algorithm have been observed for an instances with up to
1,002 cities.
Acknowledgments
Dr. Didier El Baz thanks NVIDIA for support through
Academic Partnership.
E. Frameworks for the design of combinatorial algorithms
on GPU
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