Advertisement

Journal of Heuristics

, Volume 8, Issue 4, pp 399–414 | Cite as

On the Hardness of the Quadratic Assignment Problem with Metaheuristics

  • Eric Angel
  • Vassilis Zissimopoulos
Article

Abstract

Meta-heuristics are a powerful way to approximately solve hard combinatorial optimization problems. However, for a problem, the quality of results can vary considerably from one instance to another. Understanding such a behaviour is important from a theoretical point of view, but also has practical applications such as for the generation of instances during the evaluation stage of a heuristic.

In this paper we propose a new complexity measure for the Quadratic Assignment Problem in the context of metaheuristics based on local search, e.g. simulated annealing. We show how the ruggedness coefficient previously introduced by the authors, in conjunction with the well known concept of dominance, provides important features of the search space explored during a local search algorithm, and gives a rather precise idea of the complexity of an instance for these heuristics. We comment previous experimental studies concerning tabu search methods and genetic algorithms with local search in the light of our complexity measure. New computational results with simulated annealing and taboo search are presented.

quadratic assignment problem local search complexity measures 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Angel, E. and V. Zissimopoulos. (1998). “Autocorrelation Coefficient for the Graph Bipartitioning Problem.” Theoretical Computer Science 191, 229–243.Google Scholar
  2. Angel, E. and V. Zissimopoulos. (1998). “On the Quality of Local Search for the Quadratic Assignment Problem.” Discrete Applied Mathematics 82, 15–25.Google Scholar
  3. Angel, E. and V. Zissimopoulos. (2000). “On the Classification of NP-Complete Problems in Terms of their Correlation Coefficient.” Discrete Applied Mathematics 99, 261–277.Google Scholar
  4. Angel, E. and V. Zissimopoulos. (2001). “On the Landscape Ruggedness of the Quadratic Assignment Problem.” Theoretical Computer Science 263(1/2), 159–172.Google Scholar
  5. Bachelet, V., P. Preux, and E-G. Talbi. (1996). “Parallel Hybrid Meta-Heuristics: Application to the Quadratic Assignment Problem.” Technical Report LIL-96-2, LIFL, Université des Sciences et Technologies de Lille.Google Scholar
  6. Bachelet, V., P. Preux, and E-G. Talbi. (1997). “The Landscape of the Quadratic Assignment Problem and Local Search Methods.” In Tenth Meeting of the European Chapter on Combinatorial Optimization. Teneriffe, Canary Islands.Google Scholar
  7. Block, T.E. (1979). “On the Complexity of Facilities Layout Problems.” Management Science 25, 280.Google Scholar
  8. Burkard, R.E. and U. Fincke. (1985). “Probabilistic Asymptotic Properties of Some Combinatorial Optimization Problems.” Discrete Applied Mathematics 12, 21–29.Google Scholar
  9. Burkard, R.E., S.E. Karisch, and F. Rendl. (1997). “QAPLIB—AQuadratic Assignment Problem Library.” Journal of Global Optimization 10, 391–403. See also at http://www.imm.dtu.dk/∼sk/qaplib/.Google Scholar
  10. Connolly, D.T. (1990). “An Improved Annealing Scheme for the QAP.” European Journal of Operational Research 46, 93–100.Google Scholar
  11. Dorigo, M., V. Maniezzo, and A. Colorni. (1995). Algodesk: An Experimental Comparison of Eight Evolutionary Heuristics for the Quadratic Assignment Problem.” European Journal of Operational Research 81, 188–204.Google Scholar
  12. Fonlupt, C., D. Robilliard, P. Preux, and E-G. Talbi. (1997). “Fitness Landscapes and Performance of Meta-Heuristics.” In 2nd International Conference on Metaheuristics. Sophia-Antipolis, France.Google Scholar
  13. Garey, M.R. and D.S. Johnson. (1979). Computers and Intractability—A Guide to the Theory of NP-Completeness. W.H. Freeman and Company.Google Scholar
  14. Herroelen, W. and A. Van Gils. (1985). “On the Use of Flow Dominance in Complexity Measures for Facility Layout Problems.” International Journal of Production Research 23(1), 97–108.Google Scholar
  15. Johnson, D.S., C.R. Aragon, L.A. McGeoch, and C. Schevon. (1989). “Optimization by Simulated Annealing: An Experimental Evaluation; part I, Graph Partitioning.” Operations Research 37(6), 865–892.Google Scholar
  16. Li, Y. and P.M. Pardalos. (1992). “Generating Quadratic Assignment Test Problems with Known Optimal Permutations.” Computational Optimization and Applications 1, 163–184.Google Scholar
  17. Martin O. (1998). “Propriétés des solutions engendrées par des algorithmes heuristiques: la limite de grande taille.” In Premier Congrès de la Société Française de Recherche Opérationnelle et Aide `a la Décision. Paris, Janvier.Google Scholar
  18. Mautor, T. and C. Roucairol. (1993). “Difficulties of Exact Methods for Solving the Quadratic Assignment Problem.” In DIMACS (Series in Discrete Mathematics and Theoretical Computer Science) Workshop, Vol. 16, pp. 263–274.Google Scholar
  19. Merz, P. and B. Freisleben. (1997). “A Genetic Local Search Approach to the Quadratic Assignment Problem.” In Proceedings of the Seventh International Conference on Genetic Algorithms (ICGA'97). East Lansing, USA.Google Scholar
  20. Reidys, C.M. and P.F. Stadler. “Combinatorial Landscapes.” SIAM Review, to appear.Google Scholar
  21. Sahni, S. and T. Gonzalez. (1976). “P-Complete Approximation Problems.” Journal of the ACM 23, 555–565.Google Scholar
  22. Schreiber, G.R. and O. Martin. (1999). “Cut Size Statistics of Graph Bisection Heuristics.” SIAM Journal on Optimization 20(1), 231–251.Google Scholar
  23. Stadler, P.F. (1995). “Landscapes and their Correlations Functions.” Technical Report 95-07-067, Santa Fe Institute, Santa Fe, NM.Google Scholar
  24. Taillard, E.D. (1991). “Robust Taboo Search for the QAP.” Parallel Computing 17, 443–455.Google Scholar
  25. Taillard, E.D. (1995). “Comparison of Iterative Searches for the Quadratic Assignment Problem.” Location Science 3, 87–105.Google Scholar
  26. Taillard, E.D. www.idsia.ch/∼eric.Google Scholar
  27. Vollmann, T.E. and E.S. Buffa. (1966). “The Facilities Layout Problem in Perspective.” Management Science 12(10), 450–468.Google Scholar
  28. Weinberger, E.D. (1990). “Correlated and Uncorrelated Fitness Landscapes and How to Tell the Difference.” Biological Cybernetics 63, 325–336.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Eric Angel
    • 1
  • Vassilis Zissimopoulos
    • 2
  1. 1.LaMIUniversité d'Évry-Val d'EssonneEvryFrance
  2. 2.LIPNUniversité Paris NordVilletaneuseFrance

Personalised recommendations

Cite article

Buy options