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

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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.

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References

  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

    Article  Google Scholar 

  2. Balas, E., Yu, C.S.: Finding a maximum clique in an arbitrary graph. SIAM J. Comput. 15(4), 1054–1068 (1986)

    Article  Google Scholar 

  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)

  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)

  5. Bomze, I., Budinich, M., Pardalos, P.M., Pelillo, M.: The Maximum Clique Problem. Handbook of Combinatorial Optimization. Kluwer, Dordrecht (1999)

    Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  10. Du, D., Pardalos, P.M.: Handbook of Combinatorial Optimization, Supplement vol A, p. 648. Springer, Berlin (1999)

  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)

  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

    Article  Google Scholar 

  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. Glover, F., Laguna, M.: Tabu Search. Kluwer, Dordrecht (1997)

    Book  Google Scholar 

  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

    Article  Google Scholar 

  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. Jerrum, M.: Large cliques elude the metropolis process. Random Struct. Algorithms 3(4), 347–359 (1992). doi:10.1002/rsa.3240030402

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. 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. 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)

  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)

  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)

  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)

  25. Mycielski, J.: Sur le coloriage des graphes. Colloq. Math. 3, 161–162 (1955)

    Google Scholar 

  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. 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

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

  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

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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).

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Authors and Affiliations

  1. Laboratory of Algorithms and Technologies for Networks Analysis, National Research University Higher School of Economics, 136 Rodionova Street, Niznhy Novgorod, Russia

    Evgeny Maslov, Mikhail Batsyn & Panos M. Pardalos

  2. Center of Applied Optimization, University of Florida, 303 Weil Hall, Gainesville, FL, 32611, USA

    Panos M. Pardalos

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  1. Evgeny Maslov
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  2. Mikhail Batsyn
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  3. Panos M. Pardalos
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Correspondence to Mikhail Batsyn.

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Maslov, E., Batsyn, M. & Pardalos, P.M. Speeding up branch and bound algorithms for solving the maximum clique problem. J Glob Optim 59, 1–21 (2014). https://doi.org/10.1007/s10898-013-0075-9

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