A Simple and Faster Branch-and-Bound Algorithm for Finding a Maximum Clique
This paper proposes new approximate coloring and other related techniques which markedly improve the run time of the branch-and-bound algorithm MCR (J. Global Optim., 37, 95–111, 2007), previously shown to be the fastest maximum-clique-finding algorithm for a large number of graphs. The algorithm obtained by introducing these new techniques in MCR is named MCS. It is shown that MCS is successful in reducing the search space quite efficiently with low overhead. Consequently, it is shown by extensive computational experiments that MCS is remarkably faster than MCR and other existing algorithms. It is faster than the other algorithms by an order of magnitude for several graphs. In particular, it is faster than MCR for difficult graphs of very high density and for very large and sparse graphs, even though MCS is not designed for any particular type of graphs. MCS can be faster than MCR by a factor of more than 100,000 for some extremely dense random graphs.
Unable to display preview. Download preview PDF.
- 2.Balas, E., Ceria, S., Cornuéjols, G., Pataki, G.: Polyhedral methods for the maximum clique problem. In: Johnson, Trick (eds.) , pp. 11–28 (1996)Google Scholar
- 3.Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillo, M.: The maximum clique problem. In: Du, D.-Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, Supplement vol. A, pp. 1–74 (1999) Google Scholar
- 4.Bourjolly, J.-M., Gill, P., Laporte, G., Mercure, H.: An exact quadratic 0-1 algorithm for the stable set problem. In: Johnson, Trick (eds.) , pp. 53–73 (1996)Google Scholar
- 5.Brown, J.B., Bahadur, D.K.C., Tomita, E., Akutsu, T.: Multiple methods for protein side chain packing using maximum weight cliques. Genome Inform. 17, 3–12 (2006)Google Scholar
- 9.Fujii, T., Tomita, E.: On efficient algorithms for finding a maximum clique. Technical Report of IECE, AL81-113, pp. 25–34 (1982)Google Scholar
- 10.Higashi, T., Tomita, E.: A more efficient algorithm for finding a maximum clique based on an improved approximate coloring. Technical Report of Univ. Electro-Commun, UEC-TR-CAS5-2006 (2006)Google Scholar
- 13.Johnson, D.S., Trick, M.A. (eds.): Cliques, Coloring, and Satisfiability. DIMACS Series in Discr. Math. and Theoret. Comput. Sci., vol. 26 (1996)Google Scholar
- 15.Matsunaga, T., Yonemori, C., Tomita, E., Muramatsu, M.: Clique-based data mining for related genes in a biomedical database. BMC Bioinformatics 10 (2009)Google Scholar
- 17.Régin, J.C.: Using constraint programming to solve the maximum clique problem. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 634–648. Springer, Heidelberg (2003)Google Scholar
- 20.Shindo, M., Tomita, E., Maruyama, Y.: An efficient algorithm for finding a maximum clique. Technical Report of IEC, CAS86-5, pp. 33–40 (1986)Google Scholar
- 22.Sutani, Y., Tomita, E.: Computational experiments and analyses of a more efficient algorithm for finding a maximum clique. Technical Report of IPSJ, 2005-MPS-57, pp. 45–48 (2005)Google Scholar
- 23.Tomita, E., Kameda, T.: An efficient branch-and-bound algorithm for finding a maximum clique with computational experiments. J. Global Optim. 37, 95–111 (2009); J. Global Optim. 37, 95–311 (2007)Google Scholar
- 25.Tomita, E., Tanaka, A., Takahashi, H.: The worst-case time complexity for generating all maximal cliques and computational experiments (An invited paper in the Special Issue on COCOON 2004). Theoret. Comput. Sci. 363, 28–42 (2006); The preliminary version appeared in Tomita, E., Tanaka, A., Takahashi, H.: The worst-case time complexity for generating all maximal cliques. In: Chwa, K.-Y., Munro, J.I.J. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 161–170. Springer, Heidelberg (2004)Google Scholar