Advertisement

Journal of Global Optimization

, Volume 59, Issue 1, pp 1–21 | Cite as

Speeding up branch and bound algorithms for solving the maximum clique problem

  • Evgeny Maslov
  • Mikhail BatsynEmail author
  • Panos M. Pardalos
Article

Abstract

In this paper we consider two branch and bound algorithms for the maximum clique problem which demonstrate the best performance on DIMACS instances among the existing methods. These algorithms are MCS algorithm by Tomita et al. (2010) and MAXSAT algorithm by Li and Quan (2010a, b). We suggest a general approach which allows us to speed up considerably these branch and bound algorithms on hard instances. The idea is to apply a powerful heuristic for obtaining an initial solution of high quality. This solution is then used to prune branches in the main branch and bound algorithm. For this purpose we apply ILS heuristic by Andrade et al. (J Heuristics 18(4):525–547, 2012). The best results are obtained for p_hat1000-3 instance and gen instances with up to 11,000 times speedup.

Keywords

Maximum clique problem Branch and bound algorithm  Heuristic solution Graph colouring 

Mathematics Subject Classification (2000)

05C69 05C85 90C27 90C59 90-08 

Notes

Acknowledgments

The authors would like to thank professor Mauricio Resende and his co-authors for the source code of their powerful ILS heuristic. We are also thankful to Chu-Min Li and Zhe Quan for the source code of their efficient MAXSAT algorithm. The authors are supported by LATNA Laboratory, National Research University Higher School of Economics (NRU HSE), Russian Federation government grant, ag. 11.G34.31.0057. Mikhail Batsyn is supported by Federal Grant-in-Aid Program “Research and development on priority directions of development of the scientific-technological complex of Russia for 2007–2013” (Governmental Contract No. 14.514.11.4065).

References

  1. 1.
    Andrade, D.V., Resende, M.G.C., Werneck, R.F.: Fast local search for the maximum independent set problem. J. Heuristics 18(4), 525–547 (2012). doi: 10.1007/s10732-012-9196-4 CrossRefGoogle Scholar
  2. 2.
    Balas, E., Yu, C.S.: Finding a maximum clique in an arbitrary graph. SIAM J. Comput. 15(4), 1054–1068 (1986)CrossRefGoogle Scholar
  3. 3.
    Bertoni, A., Campadelli, P., Grossi, G.: A discrete neural algorithm for the maximum clique problem: analysis and circuit implementation. In: Proceedings of Workshop on Algorithm, Engineering, WAE’97, pp. 84–91 (1997)Google Scholar
  4. 4.
    Boginski, V., Butenko, S., Pardalos, P.M.: On structural properties of the market graph. In: Nagurney, A. (ed.) Innovations in Financial and Economic Networks, pp 29–45. Edward Elgar Publishing, London (2003)Google Scholar
  5. 5.
    Bomze, I., Budinich, M., Pardalos, P.M., Pelillo, M.: The Maximum Clique Problem. Handbook of Combinatorial Optimization. Kluwer, Dordrecht (1999)Google Scholar
  6. 6.
    Bron, C., Kerbosch, J.: Algorithm 457: finding all cliques of an undirected graph. Commun. ACM 16(9), 575–577 (1973). doi: 10.1145/362342.362367 CrossRefGoogle Scholar
  7. 7.
    Brouwer, A., Shearer, J., Sloane, N., Smith, W.: A new table of constant weight codes. IEEE Trans. Inf. Theory 36(6), 1334–1380 (1990). doi: 10.1109/18.59932 CrossRefGoogle Scholar
  8. 8.
    Butenko, S., Wilhelm, W.E.: Clique-detection models in computational biochemistry and genomics. Eur. J. Oper. Res. 173(1), 1–17 (2006). doi: 10.1016/j.ejor.2005.05.026 CrossRefGoogle Scholar
  9. 9.
    Carraghan, R., Pardalos, P.M.: An exact algorithm for the maximum clique problem. Oper. Res. Lett. 9(6), 375–382 (1990). doi: 10.1016/0167-6377(90)90057-C CrossRefGoogle Scholar
  10. 10.
    Du, D., Pardalos, P.M.: Handbook of Combinatorial Optimization, Supplement vol A, p. 648. Springer, Berlin (1999)Google Scholar
  11. 11.
    Fahle, T.: Simple and fast: improving a branch-and-bound algorithm for maximum clique. In: Proceedings of the 10th Annual European Symposium on Algorithms (ESA ’02), pp 485–498. Springer, London, UK (2002)Google Scholar
  12. 12.
    Feo, T.A., Resende, M.G.C.: Greedy randomized adaptive search procedures. J. Glob. Optim. 6(2), 109–133 (1995). doi: 10.1007/BF01096763 CrossRefGoogle Scholar
  13. 13.
    Funabiki, N., Takefuji, Y., Lee, K.C.: A neural network model for finding a near-maximum clique. J. Parallel Distrib. Comput. 14(3), 340–344 (1992). doi: 10.1016/0743-7315(92)90072-U
  14. 14.
    Glover, F., Laguna, M.: Tabu Search. Kluwer, Dordrecht (1997)CrossRefGoogle Scholar
  15. 15.
    Grosso, A., Locatelli, M., Pullan, W.: Simple ingredients leading to very efficient heuristics for the maximum clique problem. J. Heuristics 14(6), 587–612 (2008). doi: 10.1007/s10732-007-9055-x CrossRefGoogle Scholar
  16. 16.
    Jenelius, E., Petersen, T., Mattsson, L.: Importance and exposure in road network vulnerability analysis. Transp. Res. Part A: Policy Pract. 40(7), 537–560 (2006). doi: 10.1016/j.tra.2005.11.003 Google Scholar
  17. 17.
    Jerrum, M.: Large cliques elude the metropolis process. Random Struct. Algorithms 3(4), 347–359 (1992). doi: 10.1002/rsa.3240030402 CrossRefGoogle Scholar
  18. 18.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972)CrossRefGoogle Scholar
  19. 19.
    Konc, J., Janezic, D.: An improved branch and bound algorithm for the maximum clique problem. MATCH Commun. Math. Comput. Chem. 58, 569–590 (2007)Google Scholar
  20. 20.
    Kopf, R., Ruhe, G.: A computational study of the weighted independent set problem for general graphs. Found. Control Eng. 12, 167–180 (1987)Google Scholar
  21. 21.
    Li, C.M., Quan, Z.: Combining graph structure exploitation and propositional reasoning for the maximum clique problem. In: Proceedings of the 2010 22nd IEEE International Conference on Tools with Artificial Intelligence—Volume 01 (ICTAI’10), pp 344–351. IEEE, Arras, France (2010a)Google Scholar
  22. 22.
    Li, C.M., Quan, Z.: An efficient branch-and-bound algorithm based on maxsat for the maximum clique problem. In: Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligence (AAAI-10), pp. 128–133. AAAI Press, Atlanta, USA (2010b)Google Scholar
  23. 23.
    Marchiori, E.: Genetic, iterated and multistart local search for the maximum clique problem. In: Applications of Evolutionary Computing, Springer, LNCS, pp. 112–121. Springer, Berlin (2002)Google Scholar
  24. 24.
    Matula, D.W., Marble, G., Isaacson, J.D.: Graph coloring algorithms. In: Read, R.C. (ed.) Graph Theory and Computing, pp 109–122. Academic Press, New York (1972)Google Scholar
  25. 25.
    Mycielski, J.: Sur le coloriage des graphes. Colloq. Math. 3, 161–162 (1955)Google Scholar
  26. 26.
    Pullan, W., Hoos, H.H.: Dynamic local search for the maximum clique problem. J. Artif. Int. Res. 25(1), 159–185 (2006)Google Scholar
  27. 27.
    Singh, A., Gupta, A.K.: A hybrid heuristic for the maximum clique problem. J. Heuristics 12(1–2), 5–22 (2006). doi: 10.1007/s10732-006-3750-x CrossRefGoogle Scholar
  28. 28.
    Sloane, N.J.A.: Unsolved problems in graph theory arising from the study of codes. Graph Theory Notes NY 18(11), 11–20 (1989)Google Scholar
  29. 29.
    Tomita, E., Kameda, T.: An efficient branch-and-bound algorithm for finding a maximum clique with computational experiments. J. Glob. Optim. 37(1), 95–111 (2007)CrossRefGoogle Scholar
  30. 30.
    Tomita, E., Seki, T.: An efficient branch-and-bound algorithm for finding a maximum clique. In: Proceedings of the 4th International Conference on Discrete Mathematics and Theoretical Computer Science (DMTCS’03), pp. 278–289. Springer, Berlin, Heidelberg (2003)Google Scholar
  31. 31.
    Tomita, E., Sutani, Y., Higashi, T., Takahashi, S., Wakatsuki, M.: A simple and faster branch-and-bound algorithm for finding a maximum clique. In: Proceedings of the 4th International Conference on Algorithms and Computation (WALCOM’10), pp. 191–203. Springer, Berlin, Heidelberg (2010). doi:  10.1007/978-3-642-11440-3_18

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Evgeny Maslov
    • 1
  • Mikhail Batsyn
    • 1
    Email author
  • Panos M. Pardalos
    • 1
    • 2
  1. 1.Laboratory of Algorithms and Technologies for Networks AnalysisNational Research University Higher School of EconomicsNiznhy NovgorodRussia
  2. 2.Center of Applied OptimizationUniversity of FloridaGainesvilleUSA

Personalised recommendations