Abstract
A greedy randomized adaptive search procedure (GRASP) is a metaheuristic for combinatorial optimization. Given an undirected graph with weights associated with its nodes, the Steiner tree problem consists in finding a minimum weight subgraph spanning a given subset of (terminal) nodes of the original graph. In this paper, we describe a parallel GRASP for the Steiner problem in graphs. We review basic concepts of GRASP: construction and local search algorithms. The implementation of a sequential GRASP for the Steiner problem in graphs is described in detail. Feasible solutions are characterized by their non-terminal nodes. A randomized version of Kruskal's algorithm for the minimum spanning tree problem is used in the construction phase. Local search is based on insertions and eliminations of nodes to/from the current solution. Parallelization is done through the distribution of the GRASP iterations among the processors on a demand-driven basis, in order to improve load balancing. The parallel procedure was implemented using the Message Passing Interface library on an IBM SP2 machine. Computational experiments on benchmark problems are reported.
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References
A.C. Alvim, Evaluation of GRASP parallelization strategies (in Portuguese), M.Sc. Dissertation, Department of Computer Science, Catholic University of Rio de Janeiro, 1998.
J.E. Beasley, “OR-Library: Distributing test problems by electronic mail”, Journal of the Operational Research Society 41 (1990), 1069–1072.
E.-A. Choukmane, “Une heuristique pour le problème de l'arbre de Steiner”, RAIRO Recherche Opérationnelle 12 (1978), 207–212.
K.A. Dowsland, “Hill-climbing simulated annealing and the Steiner problem in graphs”, Engineering Optimization 17 (1991), 91–107.
C.W. Duin and A. Volgenant, “Reduction tests for the Steiner problem in graphs”, Networks 19 (1989), 549–567.
C.W. Duin and S. Voss, “Efficient path and vertex exchange in Steiner tree algorithms”, Networks 29 (1997), 89–105.
H. Esbensen, “Computing near-optimal solutions to the Steiner problem in a graph using a genetic algorithm”, Networks 26 (1995), 173–185.
T.A. Feo and M.G. Resende, “Greedy randomized adaptive search procedures”, Journal of Global Optimization 6 (1995), 109–133.
M. Gendreau, J.-F. Larochelle and B. Sansó, “A tabu search heuristic for the Steiner tree problem in graphs”, GERAD, Rapport de recherche G-96-03, Montréal, 1996.
F.K. Hwang, D.S. Richards and P. Winter, The Steiner tree problem, NorthHolland, Amsterdam, 1992.
A. Iwainsky, E. Canuto, O. Taraszow and A. Villa, “Network decomposition for the optimization of connection structures”, Networks 16 (1986), 205–235.
A. Kapsalis, V.J. Rayward-Smith and G.D. Smith, “Solving the graphical Steiner tree problem using genetic algorithms”, Journal of the Operational Research Society44 (1993), 397–406.
R.M. Karp, “Reducibility among combinatorial problems”, in Complexity of Computer Computations (E. Miller and J.W. Thatcher, eds.), 85–103, Plenum Press, New York, 1972.
L.T. Kou G. Markowsky and L. Berman, “A fast algorithm for Steiner trees”, Acta Informatica 15 (1981), 141–145.
J.B. Kruskal, “On the shortest apanning subtree of a graph and the traveling salesman problem”, Proceedings of the American Mathematical Society 7 (1956), 48–50.
N. Maculan, “The Steiner problem in graphs”, in Surveys in Combinatorial Optimization (S. Martello, G. Laporte, M. Minoux, and C.C. Ribeiro, eds.), Annals of Discrete Mathematics 31 (1987), 185–212.
S.L. Martins, P. Pardalos, M.G. Resende. and C.C. Ribeiro, “GRASP procedures for the Steiner problem in graphs”, presented at the DIMACS Workshop on Randomization Methods in Algorithm Design, research report in preparation, 1997.
K. Mehlhorn, “A faster approximation for the Steiner problem in graphs”, Information Processing Letters 27 (1988), 125–128.
M. Minoux, “Efficient greedy heuristics for Steiner tree problems using reoptimization and supermodularity”, INFOR 28 (1990), 221–233.
Message Passing Interface Forum, “MPI: A new message-passing interface standard (version 1.1)”, Technical report, University of Tennessee, Knoxville, 1995.
J. Plesník, “A bound for the Steiner problem in graphs”, Math. Slovaca 31 (1981), 155–163.
M. Prais and C.C. Ribeiro, “Reactive GRASP: An application to a matrix decomposition problem in TDMA traffic assignment”, Research paper submitted for publication, Catholic University of Rio de Janeiro, Department of Computer Science, 1998.
M.G. Resende and C.C. Ribeiro, “A GRASP for Graph Planarization”, Networks 29 (1997), 173–189.
C.C. Ribeiro and M.C. Souza, “An improved tabu search for the Steiner problem in graphs”, Working paper, Catholic University of Rio de Janeiro, Department of Computer Science, 1997.
S. Voss, “Steiner's problem in graphs: Heuristic methods”, Discrete Applied Mathematics 40 (1992), 45–72.
D.W. Walker, “The design of a standard message passing interface for distributed memory concurrent computers”, Parallel Computing 20 (1994), 657–673.
P. Winter, “Steiner problem in networks: A survey”, Networks 17 (1987), 129–167.
J. Xu, S.Y. Chiu and F. Glover, “Tabu search heuristics for designing a Steiner tree based digital line network”, Working paper, University of Colorado at Boulder, 1995.
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Martins, S.L., Ribeiro, C.C., Souza, M.C. (1998). A parallel GRASP for the Steiner problem in graphs. In: Ferreira, A., Rolim, J., Simon, H., Teng, SH. (eds) Solving Irregularly Structured Problems in Parallel. IRREGULAR 1998. Lecture Notes in Computer Science, vol 1457. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0018547
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DOI: https://doi.org/10.1007/BFb0018547
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