Abstract
A GRASP (greedy randomized adaptive search procedure) is a multi-start metaheuristic for combinatorial optimization. We study the probability distributions of solution time to a sub-optimal target value in five GRASPs that have appeared in the literature and for which source code is available. The distributions are estimated by running 12,000 independent runs of the heuristic. Standard methodology for graphical analysis is used to compare the empirical and theoretical distributions and estimate the parameters of the distributions. We conclude that the solution time to a sub-optimal target value fits a two-parameter exponential distribution. Hence, it is possible to approximately achieve linear speed-up by implementing GRASP in parallel.
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References
Battiti, R. and G. Tecchiolli. (1992). “Parallel Biased Search for Combinatorial Optimization: Genetic Algorithms and TABU.” Microprocessors and Microsystems 16, 351–367.
Bollobás, B. (1985). Random Graphs. San Diego, CA: Academic Press.
Burkard, R., S. Karisch, and F. Rendl. (1991). “QAPLIB—A Quadratic Assignment Problem Library.” European Journal of Operations Research 55, 115–119.
Chambers, J.M., W.S. Cleveland, B. Kleiner, and P.A. Tukey. (1983). Graphical Methods for Data Analysis. London: Chapman &;; Hall.
Cimikowski, R. (1995). “An Analysis of Heuristics for the Maximum Planar Subgraph Problem.” In Proceedings of the 6th ACM-SIAM Symposium on Discrete Algorithms, pp. 322–331.
Dodd, N. (1990). “Slow Annealing versus Multiple Fast Annealing Runs: An Empirical Investigation.” Parallel Computing 16, 269–272.
Eikelder, H.T., M. Verhoeven, T. Vossen, and E. Aarts. (1996). “A Probabilistic Analysis of Local Search.” In I. Osman and J. Kelly (eds.), Metaheuristics: Theory &;; Applications. Norwell, MA: Kluwer Academic Publishers, pp. 605–618.
Feo, T. and M. Resende. (1989). “A Probabilistic Heuristic for a Computationally Difficult Set Covering Problem.” Operations Research Letters 8, 67–71.
Feo, T. and M. Resende. (1995). “Greedy Randomized Adaptive Search Procedures.” Journal of Global Optimization 6, 109–133.
Feo, T., M. Resende, and S. Smith. (1994). “A Greedy Randomized Adaptive Search Procedure for Maximum Independent Set.” Operations Research 42, 860–878.
Festa, P. and M. Resende. (2000). “GRASP: An Annotated Bibliography.” Technical Report, AT &;; T Labs Research, Florham Park, NJ 07733.
Fleurent, C. and F. Glover. (1999). “Improved Constructive Multistart Strategies for the Quadratic Assignment Problem using Adaptive Memory.” INFORMS Journal on Computing 11, 198–204.
Goldschmidt, O. and A. Takvorian. (1994). “An Efficient Graph Planarization Two-Phase Heuristic.” Networks 24, 69–73.
Hart, J. and A. Shogan. (1987). “Semi-Greedy Heuristics: An Empirical Study.” Operations Research Letters 6, 107–114.
Hoos, H. and T. Stützle. (1999). “Towards a Characterisation of the Behaviour of Stochastic Local Search Algorithms for SAT.” Artificial Intelligence 112, 213–232.
Li, Y., P. Pardalos, and M. Resende. (1994). “A Greedy Randomized Adaptive Search Procedure for the Quadratic Assignment Problem.” In P. Pardalos and H. Wolkowicz (eds.), Quadratic Assignment and Related Problems, Vol. 16 of DIMACS Series on Discrete Mathematics and Theoretical Computer Science. Providence, RI: American Mathematical Society, pp. 237–261.
Martins, S., M. Resende, C. Ribeiro, and P. Pardalos. (2000). “A Parallel GRASP for the Steiner Tree Problem in Graphs Using a Hybrid Local Search Strategy.” Journal of Global Optimization 17, 267–283.
Martins, S., C. Ribeiro, and M. Souza. (1998). “A Parallel GRASP for the Steiner Problem in Graphs.” In A. Ferreira and J. Rolim (eds.), Proceedings of IRREGULAR'98–5th International Symposium on Solving Irregularly Structured Problems in Parallel, Vol. 1457 of Lecture Notes in Computer Science. Berlin: Springer-Verlag, pp. 285–297.
Murphey, R., P. Pardalos, and L. Pitsoulis. (1998). “A Parallel GRASP for the Data Association Multidimensional Assignment Problem.” In P. Pardalos (ed.), Parallel Processing of Discrete Problems, Vol. 106 of The IMA Volumes in Mathematics and Its Applications. Berlin: Springer-Verlag, pp. 159–180.
Osborne, L. and B. Gillett. (1991). “A Comparison of Two Simulated Annealing Algorithms Applied to the Directed Steiner Problem on Networks.” In ORSA J. Computing 3, 213–225.
Pardalos, P., L. Pitsoulis, and M. Resende. (1995). “A Parallel GRASP Implementation for the Quadratic Assignment Problem.” In A. Ferreira and J. Rolim (eds.), Parallel Algorithms for Irregularly Structured Problems— Irregular'94. Norwell, MA: Kluwer Academic Publishers, pp. 111–130.
Pardalos, P., L. Pitsoulis, and M. Resende. (1996). “A Parallel GRASP for MAX-SAT Problems.” In Lecture Notes in Computer Science, Vol. 1184, pp. 575–585.
Pardalos, P., L. Pitsoulis, and M. Resende. (1997). “Algorithm 769 Fortran Subroutines for Approximate Solution of Sparse Quadratic Assignment Problems using GRASP.” ACM Transactions on Mathematical Software 23, 196–208.
Rangel, M., N. Abreu, and P. Boaventura Netto. (1999) “GRASP: In the QAP: An Acceptance Bound for Initial Solution.” In Proc. of the Third Metaheuristics International Conference, pp. 381–386.
Rangel, M., N. de Abreu, P. Boaventura Netto, and M. Boeres. (1998), “A Modified Local Search for GRASP in the Quadratic Assignment Problem.” Technical Report, Production Engineering Program, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, RJ Brazil.
Resende, M. (1998). “Computing Approximate Solutions of the Maximum Covering Problem using GRASP.” J. Heuristics 4, 161–171.
Resende, M. and T. Feo. (1996). “A GRASP for Satisfiability.” In D. Johnson and M. Trick (eds.), Cliques, Coloring, and Satisfiability: The Second DIMACS Implementation Challenge, Vol. 26 of DIMACS Series on Discrete Mathematics and Theoretical Computer Science. Providence, RI: American Mathematical Society, pp. 499–520.
Resende, M., T. Feo, and S. Smith. (1998). “Algorithm 787: Fortran Subroutines for Approximate Solution of Maximum Independent Set Problems using GRASP.” ACM Trans. Math. Software 24, 386–394.
Resende, M., P. Pardalos, and Y. Li. (1996). “Algorithm 754: Fortran Subroutines for Approximate Solution of Dense Quadratic Assignment Problems using GRASP.” ACM Transactions on Mathematical Software 22, 104–118.
Resende, M., L. Pitsoulis, and P. Pardalos. (1997). “Approximate Solution of Weighted MAX-SAT Problems using GRASP.” In J. Gu and P. Pardalos (eds.), Satisfiability Problems, Vol. 35 of DIMACS Series on Discrete Mathematics and Theoretical Computer Science. Providence, RI: American Mathematical Society, pp. 393–405.
Resende, M., L. Pitsoulis, and P. Pardalos. (2000). “Fortran Subroutines for Computing Approximate Solutions of MAX-SAT Problems using GRASP.” Discrete Applied Mathematics 100, 95–113.
Resende, M. and C. Ribeiro. (1997). “A GRASP for Graph Planarization.” Networks 29, 173–189.
Ribeiro, C. and M. Resende. (1999). “Algorithm 797: Fortran Subroutines for Approximate Solution of Graph Planarization Problems using GRASP.” ACM Transactions on Mathematical Software 25, 341–352.
Selman, B., H. Kautz, and B. Cohen. (1994). “Noise Strategies for Improving Local Search.” In Proceedings of the AAAI-94, Cambridge, MA: MIT Press, pp. 337–343.
Taillard, E. (1991). “Robust Taboo Search for the Quadratic Assignment Problem.” Parallel Computing 17, 443–455.
Verhoeven, M. and E. Aarts. (1995). “Parallel Local Search.” J. Heuristics 1, 43–66.
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Aiex, R.M., Resende, M.G. & Ribeiro, C.C. Probability Distribution of Solution Time in GRASP: An Experimental Investigation. Journal of Heuristics 8, 343–373 (2002). https://doi.org/10.1023/A:1015061802659
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DOI: https://doi.org/10.1023/A:1015061802659