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Solving the maximum clique problem using a tabu search approach

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Abstract

We describe two variants of a tabu search heuristic, a deterministic one and a probabilistic one, for the maximum clique problem. This heuristic may be viewed as a natural alternative implementation of tabu search for this problem when compared to existing ones. We also present a new random graph generator, the\(\hat p\)-generator, which produces graphs with larger clique sizes than comparable ones obtained by classical random graph generating techniques. Computational results on a large set of test problems randomly generated with this new generator are reported and compared with those of other approximate methods.

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References

  1. G. Avondo-Bodeno,Economic Applications of the Theory of Graphs (Gordon and Breach, New York, 1962).

    Google Scholar 

  2. L. Babel, Finding maximum cliques in arbitrary and in special graphs, Report TUM-M9008, Mathematisches Institut und Institut für Informatik, Technische Universität München (1990), to appear in Computing.

  3. L. Babel and G. Tinhofer, A branch and bound algorithm for the maximum clique problem, Zeits. Oper. Res. 34(1990)207–217.

    Google Scholar 

  4. E. Balas and J. Xue, Minimum weighted coloring of triangulated graphs, with application to maximum weight vertex packing and clique finding in arbitrary graphs, SIAM J. Comput. 20(1991)209–221.

    Google Scholar 

  5. E. Balas and C.S Yu, Finding a maximum clique in an arbitrary graph, SIAM J. Comput. 15(1986)1054–1068.

    Google Scholar 

  6. C. Berge,Théorie des Graphes et ses Applications (Dunod, Paris, 1962).

    Google Scholar 

  7. B. Bollobas,Random Graphs (Academic Press, London, 1985).

    Google Scholar 

  8. B. Bollobas and P. Erdös, Cliques in random graphs, Math. Proc. Camb. Phil. Soc. 80(1976)419–427.

    Google Scholar 

  9. C. Bron and J. Kerbosch, Finding all cliques of an undirected graph, Comm. ACM 16(1973)575–577.

    Google Scholar 

  10. R. Carraghan and P.M. Pardalos, An exact algorithm for the maximum clique problem, Oper. Res. Lett. 9(1990)375–382.

    Google Scholar 

  11. V. Degot and J.M. Hualde, De l'utilisation de la notion de clique (sous-graphe complet symétrique) en matière de typologie des populations, Revue française d'automatique et recherche opérationelle: Recherche opérationelle 9(1975)5–18.

    Google Scholar 

  12. N. Deo,Graph Theory with Applications to Engineering and Computer Science (Prentice-Hall, Englewood Cliffs, 1974).

    Google Scholar 

  13. C. Ebenegger, P.L. Hammer and D. de Werra, Pseudo-Boolean functions and stability of graphs, Ann. Discr. Math. 19(1984)83–98.

    Google Scholar 

  14. C. Friden, A Hertz and D. de Werra, Stabulus: a technique for finding stable sets in large graphs with tabu search, Computing 42(1989)35–44.

    Google Scholar 

  15. C. Friden, A Hertz and D. de Werra, Tabaris: An exact algorithm based on tabu search for finding a maximum independent set in a graph, Comput. Oper. Res. 17(1990)437–445.

    Google Scholar 

  16. M.R. Garey and D.S. Johnson,Computers and Intractibility: a Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979).

    Google Scholar 

  17. M. Gendreau, A fast greedy algorithm for the maximum clique problem, paper presented at the TIMS/ORSA Meeting, New Orleans (May 1987).

  18. M. Gendreau, J.-C. Picard and L. Zubieta, An efficient implicit enumeration algorithm for the maximum clique problem, in:Advances in Optimization and Control ed. H.A. Eiselt and G. Pederzoli (Springer, Berlin, 1988).

    Google Scholar 

  19. M. Gendreau, L. Salvail and P. Soriano, An appraisal of greedy heuristics for the maximum clique problem, Centre de Recherche sur les Transports, Université de Montréal, forthcoming.

  20. M. Gendreau, P. Soriano and L. Salvail, Simulated annealing and cliques, Centre de Recherche sur les Transports, Université de Montréal, forthcoming; paper presented at the ORSA/TIMS Meeting, New York (October 1989).

  21. L. Gerhards and W. Lindenberg, Clique detection for nondirected graphs: two new algorithms, Computing 21 (1979)295–322.

    Google Scholar 

  22. F. Glover, Future paths for integer programming and links to artificial intelligence, Comput. Oper. Res. 13(1986)533–549.

    Google Scholar 

  23. F. Glover, Tabu search. Part I, ORSA J. Comput. 1(1989)190–206.

    Google Scholar 

  24. F. Glover, Tabu search. Part II, ORSA J. Comput. 2(1990)4–32.

    Google Scholar 

  25. J.W. Greene and K.J. Supowit, Simulated annealing without rejected moves, IEEE Trans. Comput. Aided Des. CAD-5(1986)221–228.

    Google Scholar 

  26. M. Grötschel, L. Lovasz and A. Schrijver,Geometric Algorithms and Combinatorial Optimization (Springer, Berlin, 1988).

    Google Scholar 

  27. P. Hansen and B. Jaumard, Algorithms for the maximum satisfiability problem, RUTCOR Research Report 43–87, Rutgers University (1987).

  28. F. Harary, Graph theory as a structural model in the social sciences, in:Graph Theory and its Applications, ed. B. Harris (Academic Press, New York, 1970).

    Google Scholar 

  29. A. Hertz and D. de Werra, Using tabu search techniques for graph coloring, Computing 29(1987)345–351.

    Google Scholar 

  30. D.S. Johnson, Approximation algorithms for combinatorial problems, J. Comput. Syst. Sci. 9(1974)256–278.

    Google Scholar 

  31. D.S. Johnson, M. Yannakakis and C.H. Papadimitriou, On generating all maximal independent sets, Inf. Proc. Lett. 27(1988)119–123.

    Google Scholar 

  32. S. Kirkpatrick, C.D. Gelatt Jr. and M.P. Vecchi, Optimization by simulated annealing, Science 220(1983)671–680.

    Google Scholar 

  33. E. Loukakis, A new backtracking algorithm for generating the family of maximal independent sets of a graph, Comput. Math. Appl. 9(1983)583–589.

    Google Scholar 

  34. E. Loukakis and C. Tsouros, Determining the number of internal stability of a graph, Int. J. Comput. Math. 11(1982)207–220.

    Google Scholar 

  35. D.W. Matula, The employee party problem, Not. A.M.S. 19(1972)A-382.

  36. D.W. Matula, The largest clique in a random graph, Technical Report CS7608, Southern Methodist University (1976).

  37. D.W. Matula, Expose-and-merge exploration and the chromatic number of a random graph, Combinatorica 7(1987)275–284.

    Google Scholar 

  38. P.M. Pardalos and N. Desai, An algorithm for finding a maximum weighted independent set in arbitrary graph, Int. J. Comput. Math. 38(1991)163–175.

    Google Scholar 

  39. P.M. Pardalos and A. Phillips, A global optimization approach for solving the maximum clique problem, Int. J. Comput. Math. 33((1990)209–216.

    Google Scholar 

  40. P.M. Pardalos and G.P. Rodgers, A branch and bound algorithm for the maximum clique problem, Comp. Oper. Res. 19(1992)363–375.

    Google Scholar 

  41. J.M. Robson, Algorithms for the maximum independent sets, J. Algor. 7(1986)425–440.

    Google Scholar 

  42. B. Roy,Algèbre Moderne et Théorie des Graphes, Vol. 1 (Dunod, Paris, 1969).

    Google Scholar 

  43. C.E. Shannon, The zero-error capacity of a noisy channel,Symp. on Information Theory, I.R.E. Trans. 3(1956).

  44. R.E. Tarjan and A.E. Trojanowski, Finding a maximum independent set, SIAM J. Comput. 6(1977)537–546.

    Google Scholar 

  45. J. Turner and W.H. Kautz, A survey of progress in graph theory in the Soviet Union, SIAM 12(1970).

  46. D. de Werra and A. Hertz, Tabu search techniques: A tutorial and application to neural networks, OR Spektrum 11(1989)131–141.

    Google Scholar 

  47. M. Widmer and A. Hertz, A new heuristic method for solving the flow shop sequencing problem, Eur. J. Oper. Res. 41(1989)186–193.

    Google Scholar 

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Authors and Affiliations

  1. Centre de Recherche sur les Transports, Université de Montréal, succursale A, C.P. 6128, H3C 3J7, Montréal, Québec, Canada

    Michel Gendreau, Patrick Soriano & Louis Salvail

Authors
  1. Michel Gendreau
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  2. Patrick Soriano
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  3. Louis Salvail
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Additional information

The authors are grateful to the Quebec Government (Fonds F.C.A.R.) and to the Canadian Natural Sciences and Engineering Research Council (grant 0GP0038816) for financial support.

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Gendreau, M., Soriano, P. & Salvail, L. Solving the maximum clique problem using a tabu search approach. Ann Oper Res 41, 385–403 (1993). https://doi.org/10.1007/BF02023002

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