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Where the Really Hard Quadratic Assignment Problems Are: The QAP-SAT Instances

  • Conference paper
Evolutionary Computation in Combinatorial Optimization (EvoCOP 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14632))

Abstract

The Quadratic Assignment Problem (QAP) is one of the major domains in the field of evolutionary computation, and more widely in combinatorial optimization. This paper studies the phase transition of the QAP, which can be described as a dramatic change in the problem’s computational complexity and satisfiability, within a narrow range of the problem parameters. To approach this phenomenon, we introduce a new QAP-SAT design of the initial problem based on submodularity to capture its difficulty with new features. This decomposition is studied experimentally using branch-and-bound and tabu search solvers. A phase transition parameter is then proposed. The critical parameter of phase transition satisfaction and that of the solving effort are shown to be highly correlated for tabu search, thus allowing the prediction of difficult instances.

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Notes

  1. 1.

    https://github.com/afcsilva/PMITS-for-QAPVar.

  2. 2.

    https://sites.google.com/site/masakazukojima1/softwares-developed/newtbracket?pli=1.

  3. 3.

    http://mistic.heig-vd.ch/taillard/codes.dir/tabou_qap2.c.

References

  1. Achary, T., Pillay, S., Pillai, S.M., Mqadi, M., Genders, E., Ezugwu, A.E.: A performance study of meta-heuristic approaches for quadratic assignment problem. Concurr. Comput.: Pract. Experience 33(17), e6321 (2021)

    Article  Google Scholar 

  2. Ahuja, R.K., Orlin, J.B., Tiwari, A.: A greedy genetic algorithm for the quadratic assignment problem. Comput. Oper. Res. 27(10), 917–934 (2000)

    Article  MathSciNet  Google Scholar 

  3. Benlic, U., Hao, J.K.: Memetic search for the quadratic assignment problem. Expert Syst. Appl. 42(1), 584–595 (2015)

    Article  Google Scholar 

  4. Biroli, G., Cocco, S., Monasson, R.: Phase transitions and complexity in computer science: an overview of the statistical physics approach to the random satisfiability problem. Phys. A 306, 381–394 (2002)

    Article  Google Scholar 

  5. Buhmann, J.M., Dumazert, J., Gronskiy, A., Szpankowski, W.: Phase transitions in parameter rich optimization problems. In: 2017 Proceedings of the Fourteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), pp. 148–155. SIAM (2017)

    Google Scholar 

  6. Burkard, R.E., Karisch, S.E., Rendl, F.: QAPLIB-a quadratic assignment problem library. J. Global Optim. 10, 391–403 (1997)

    Article  MathSciNet  Google Scholar 

  7. Cheeseman, P.C., Kanefsky, B., Taylor, W.M.: Where the really hard problems are. In: IJCAI, vol. 91, pp. 331–337 (1991)

    Google Scholar 

  8. Chicano, F., Luque, G., Alba, E.: Elementary landscape decomposition of the quadratic assignment problem. In: Proceedings of the 12th annual conference on Genetic and evolutionary computation, pp. 1425–1432 (2010)

    Google Scholar 

  9. Commander, C.W.: A survey of the quadratic assignment problem, with applications (2005)

    Google Scholar 

  10. Daolio, F., Verel, S., Ochoa, G., Tomassini, M.: Local optima networks of the quadratic assignment problem. In: IEEE Congress on Evolutionary Computation, pp. 1–8. IEEE (2010)

    Google Scholar 

  11. Drezner, Z., Hahn, P.M., Taillard, É.D.: Recent advances for the quadratic assignment problem with special emphasis on instances that are difficult for meta-heuristic methods. Ann. Oper. Res. 139, 65–94 (2005)

    Article  MathSciNet  Google Scholar 

  12. Elorza, A., Hernando, L., Lozano, J.A.: Characterizing permutation-based combinatorial optimization problems in fourier space. Evol. Comput., 1–39 (2022)

    Google Scholar 

  13. Fujii, K., Ito, N., Kim, S., Kojima, M., Shinano, Y., Toh, K.C.: Solving challenging large scale QAPs. arXiv preprint: arXiv:2101.09629 (2021)

  14. Gent, I.P., Walsh, T.: The sat phase transition. In: ECAI, vol. 94, pp. 105–109. PITMAN (1994)

    Google Scholar 

  15. Gent, I.P., Walsh, T.: The TSP phase transition. Artif. Intell. 88(1–2), 349–358 (1996)

    Article  MathSciNet  Google Scholar 

  16. Gilmore, P.C.: Optimal and suboptimal algorithms for the quadratic assignment problem. J. Soc. Ind. Appl. Math. 10(2), 305–313 (1962)

    Article  MathSciNet  Google Scholar 

  17. Hartmann, A.K., Weigt, M.: Phase Transitions in Combinatorial Optimization Problems: Basics, Algorithms and Statistical Mechanics. John Wiley & Sons, Hoboken (2006)

    Google Scholar 

  18. Kondor, R.: A fourier space algorithm for solving quadratic assignment problems. In: Proceedings of the Twenty-first Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1017–1028. SIAM (2010)

    Google Scholar 

  19. Koopmans, T.C., Beckmann, M.: Assignment problems and the location of economic activities. Econometrica: J. Econometric Soc., 53–76 (1957)

    Google Scholar 

  20. Laurent, M., Seminaroti, M.: The quadratic assignment problem is easy for Robinsonian matrices with Toeplitz structure. Oper. Res. Lett. 43(1), 103–109 (2015)

    Article  MathSciNet  Google Scholar 

  21. Leyton-Brown, K., Hoos, H.H., Hutter, F., Xu, L.: Understanding the empirical hardness of NP-complete problems. Commun. ACM 57(5), 98–107 (2014)

    Article  Google Scholar 

  22. Loiola, E.M., De Abreu, N.M.M., Boaventura-Netto, P.O., Hahn, P., Querido, T.: A survey for the quadratic assignment problem. Eur. J. Oper. Res. 176(2), 657–690 (2007)

    Article  MathSciNet  Google Scholar 

  23. Merz, P., Freisleben, B.: Fitness landscape analysis and memetic algorithms for the quadratic assignment problem. IEEE Trans. Evol. Comput. 4(4), 337–352 (2000)

    Article  Google Scholar 

  24. Monasson, R.: Introduction to phase transitions in random optimization problems. arXiv preprint: arXiv:0704.2536 (2007)

  25. Monasson, R., Zecchina, R., Kirkpatrick, S., Selman, B., Troyansky, L.: Determining computational complexity from characteristic? Phase transitions? Nature 400(6740), 133–137 (1999)

    Article  MathSciNet  Google Scholar 

  26. Pitzer, E., Beham, A., Affenzeller, M.: Automatic algorithm selection for the quadratic assignment problem using fitness landscape analysis. In: Middendorf, M., Blum, C. (eds.) Evolutionary Computation in Combinatorial Optimization. Lecture Notes in Computer Science, vol. 7832, pp. 109–120. Springer, Berlin (2013). https://doi.org/10.1007/978-3-642-37198-1_10

    Chapter  Google Scholar 

  27. Silva, A., Coelho, L.C., Darvish, M.: Quadratic assignment problem variants: a survey and an effective parallel memetic iterated TABU search. Eur. J. Oper. Res. 292(3), 1066–1084 (2021)

    Article  MathSciNet  Google Scholar 

  28. Smith-Miles, K., Lopes, L.: Measuring instance difficulty for combinatorial optimization problems. Comput. Oper. Res. 39(5), 875–889 (2012)

    Article  MathSciNet  Google Scholar 

  29. Smith-Miles, K.A.: Towards insightful algorithm selection for optimisation using meta-learning concepts. In: 2008 IEEE International Joint Conference on Neural Networks (IEEE World Congress on Computational Intelligence), pp. 4118–4124. IEEE (2008)

    Google Scholar 

  30. Stützle, T., Fernandes, S.: New benchmark instances for the QAP and the experimental analysis of algorithms. In: Gottlieb, J., Raidl, G.R. (eds.) Evolutionary Computation in Combinatorial Optimization. Lecture Notes in Computer Science, vol. 3004, pp. 199–209. Springer, Berlin (2004). https://doi.org/10.1007/978-3-540-24652-7_20

    Chapter  Google Scholar 

  31. Taillard, É.: Robust taboo search for the quadratic assignment problem. Parallel Comput. 17(4–5), 443–455 (1991)

    Article  MathSciNet  Google Scholar 

  32. Taillard, E.D.: Comparison of iterative searches for the quadratic assignment problem. Locat. Sci. 3(2), 87–105 (1995)

    Article  Google Scholar 

  33. Tayarani-N, M.H., Prügel-Bennett, A.: Quadratic assignment problem: a landscape analysis. Evol. Intel. 8, 165–184 (2015)

    Article  Google Scholar 

  34. Verel, S., Daolio, F., Ochoa, G., Tomassini, M.: Sampling local optima networks of large combinatorial search spaces: the QAP case. In: Auger, A., Fonseca, C., Lourenc, N., Machado, P., Paquete, L., Whitley, D. (eds.) Parallel Problem Solving from Nature - PPSN XV. Lecture Notes in Computer Science(), vol. 11102, pp. 257–268. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-99259-4_21

  35. Vollmann, T.E., Buffa, E.S.: The facilities layout problem in perspective. Manage. Sci. 12(10), B–450 (1966)

    Google Scholar 

  36. Weigt, M., Hartmann, A.K.: Number of guards needed by a museum: a phase transition in vertex covering of random graphs. Phys. Rev. Lett. 84(26), 6118 (2000)

    Article  Google Scholar 

  37. Yadav, N., Murawski, C., Sardina, S., Bossaerts, P.: Phase transition in the knapsack problem. arXiv preprint: arXiv:1806.10244 (2018)

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Authors and Affiliations

  1. LISIC, Université du Littoral Côte d’Opale (ULCO), Boulogne-sur-Mer, France

    Sébastien Verel & Omar Rifki

  2. Edinburgh Napier University, Scotland, UK

    Sarah L. Thomson

Authors
  1. Sébastien Verel
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  2. Sarah L. Thomson
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  3. Omar Rifki
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Correspondence to Sébastien Verel .

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Editors and Affiliations

  1. IRIDIA, Université libre de Bruxelles, Brussels, Belgium

    Thomas Stützle

  2. Monash University, Clayton, VIC, Australia

    Markus Wagner

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Verel, S., Thomson, S.L., Rifki, O. (2024). Where the Really Hard Quadratic Assignment Problems Are: The QAP-SAT Instances. In: Stützle, T., Wagner, M. (eds) Evolutionary Computation in Combinatorial Optimization. EvoCOP 2024. Lecture Notes in Computer Science, vol 14632. Springer, Cham. https://doi.org/10.1007/978-3-031-57712-3_9

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  • DOI: https://doi.org/10.1007/978-3-031-57712-3_9

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