A New Approach for Solving the Maximum Clique Problem

  • P. J. Taillon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4041)


We describe an improved algorithm for solving the Maximum Clique problem in a graph using a novel sampling technique combined with a parameterized k-vertex cover algorithm. Experimental research shows that this approach greatly improves the execution time of the search, and in addition, provides intermediate results during computation. We also examine a very effective heuristic for finding a large clique that combines our sampling approach with fast independent set approximation. In experiments using the DIMACS benchmark, the heuristical approach established new lower bounds for four instances and provides the first optimal solution for an instance unsolved until now. The heuristic competitively matched the accuracy of the current best exact algorithm in terms of correct solutions, while requiring a fraction of the run time. Ideally such an approach could be beneficial as a preprocessing step to any exact algorithm, providing an accurate lower bound on the maximum clique, in very short time.


Exact Algorithm Vertex Cover Maximum Clique Maximum Clique Problem Minimum Vertex Cover 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Abu-Khzam, F.N., Langston, M.A., Shanbhag, P.: Scalable parallel algorithms for difficult combinatorial problems: A case study in optimization. In: Proceedings of the International Conference on Parallel and Distributed Computing and Systems (November 2003)Google Scholar
  2. 2.
    Abu-Khzam, F.N., Collins, R.L., Fellows, M.R., Langston, M.A., Suters, W.H., Symons, C.T.: Kernelization algorithms for the vertex cover problem: Theory and experiments. In: Proceedings of the ACM-SIAM Workshop on Algorithm Engineering and Experiments (January 2004)Google Scholar
  3. 3.
    Balas, E., Xue, J.: Weighted and unweighted maximum clique algorithms with upper bounds from fractional coloring. Algorithmica 15, 397–412 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Baldwin, N.E., Collins, R.L., Leuze, M.R., Langston, M.A., Symons, C.T., Voy, B.H.: High-performance computational tools for motif discovery. In: Proceedings of the IEEE International Workshop on High Performance Computational Biology (April 2004)Google Scholar
  5. 5.
    Bar-Yehuda, R., Dabholkar, V., Govindarajan, K., Sivakumar, D.: Randomized local approximations with applications to the MAX-CLIQUE problem. Technical Report 93-30, University at Buffalo (1993)Google Scholar
  6. 6.
    Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillo, M.: The maximum clique problem. In: Du, Pardalos (eds.) Handbook of Combinatorial Optimization, vol. A, pp. 1–74. Kluwer, Dordrecht (1999)Google Scholar
  7. 7.
    Cheetham, J., Dehne, F., Rau-Chaplin, A., Stege, U., Taillon, P.J.: Solving large FPT problems on coarse grained parallel machines. Journal of Computer and System Sciences 67(4), 691–706 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cheetham, J., Dehne, F., Rau-Chaplin, A., Stege, U., Taillon, P.J.: A parallel FPT application for clusters. In: Proceedings of the 3rd IEEE/ACM International Symposium on Cluster Computing and the Grid (CCGrid 2003), Tokyo, Japan, pp. 70–77 (2003)Google Scholar
  9. 9.
    Chen, J., Kanj, I.A., Jia, W.: Vertex cover: Further observations and further improvements. In: Widmayer, P., Neyer, G., Eidenbenz, S. (eds.) WG 1999. LNCS, vol. 1665, pp. 313–324. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  10. 10.
    DIMACS clique benchmarks (1993),
  11. 11.
    Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness. Congressus Numerantium 87, 161–187 (1992)MathSciNetGoogle Scholar
  12. 12.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  13. 13.
    Downey, R.G., Fellows, M.R., Regan, K.W.: Parameterized Circuit Complexity and the W Hierarchy. Theoretical Computer Science A 191, 91–115 (1998)MathSciNetGoogle Scholar
  14. 14.
    Fahle, T.: Simple and fast: Improving a branch-and-bound algorithm for maximum clique. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 485–498. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. 15.
    Focacci, F., Lodi, A., Milano, M.: Cost-based domain filtering. In: Jaffar, J. (ed.) CP 1999. LNCS, vol. 1713, pp. 189–203. Springer, Heidelberg (1999)Google Scholar
  16. 16.
    Johnson, D.S.: Approximation algorithms for combinatorial problems. Journal of Computing and System Science 9, 256–278 (1974)zbMATHCrossRefGoogle Scholar
  17. 17.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, Thatcher (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)Google Scholar
  18. 18.
    Nešetřil, J., Poljak, S.: On the complexity of the subgraph problem. Commentationes Mathematicae Universitatis Carolinae 26, 415–419 (1985)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Östegard, P.R.J.: A new algorithm for the maximum-weight clique problem. Nordic Journal of Computing 8(4), 424–436 (2001)MathSciNetGoogle Scholar
  20. 20.
    Östegard, P.R.J.: Private communication (2004)Google Scholar
  21. 21.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  22. 22.
    Pardalos, P.M., Xue, J.: The maximum clique problem. Journal of Global Optim. 4, 301–328 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Pelillo, M.: Heuristics for maximum clique and independent set. In: Floudas, Pardalos (eds.) Encyclopedia of Optimization, vol. 2, pp. 411–423. Kluwer Academic, Dordrecht (2001)Google Scholar
  24. 24.
    Regin, J.-C.: Solving the maximum clique problem with constraint programming. In: Fifth International Workshop on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems (2003)Google Scholar
  25. 25.
    Robson, J.M.: Finding a maximum independent set in time O(2n/4). Technical Report 1251-01, LaBRI, Université Bordeaux I (2001)Google Scholar
  26. 26.
    Shor, N.Z.: Dual quadratic estimates in polynomial and Boolean programming. In: Pardalos, Rosen (eds.) Computational Methods in Global Optimization. Ann. Oper. Res., vol. 25, pp. 163–168 (1990)Google Scholar
  27. 27.
    Wood, D.R.: An algorithm for finding maximum cliques in a graph. Operations Research Letters 21, 211–217 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Xue, J.: Fast Algorithms For Vertex Packing and Related Problems. Ph.D. Thesis, GSIA, Carnegie Mellon University (1991)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • P. J. Taillon
    • 1
  1. 1.School of Computer ScienceCarleton UniversityOttawaCanada

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