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A New Approach for Solving the Maximum Clique Problem

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Algorithmic Aspects in Information and Management (AAIM 2006)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 4041))

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Abstract

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.

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References

  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. 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. Balas, E., Xue, J.: Weighted and unweighted maximum clique algorithms with upper bounds from fractional coloring. Algorithmica 15, 397–412 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  10. DIMACS clique benchmarks (1993), ftp://dimacs.rutgers.edu/pub/challenge/graph/

  11. Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness. Congressus Numerantium 87, 161–187 (1992)

    MathSciNet  Google Scholar 

  12. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1998)

    MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Johnson, D.S.: Approximation algorithms for combinatorial problems. Journal of Computing and System Science 9, 256–278 (1974)

    Article  MATH  Google Scholar 

  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. Nešetřil, J., Poljak, S.: On the complexity of the subgraph problem. Commentationes Mathematicae Universitatis Carolinae 26, 415–419 (1985)

    MATH  MathSciNet  Google Scholar 

  19. Östegard, P.R.J.: A new algorithm for the maximum-weight clique problem. Nordic Journal of Computing 8(4), 424–436 (2001)

    MathSciNet  Google Scholar 

  20. Östegard, P.R.J.: Private communication (2004)

    Google Scholar 

  21. Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)

    MATH  Google Scholar 

  22. Pardalos, P.M., Xue, J.: The maximum clique problem. Journal of Global Optim. 4, 301–328 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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. 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. Wood, D.R.: An algorithm for finding maximum cliques in a graph. Operations Research Letters 21, 211–217 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  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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Author information

Authors and Affiliations

  1. School of Computer Science, Carleton University, Ottawa, Ontario, Canada

    P. J. Taillon

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  1. P. J. Taillon
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Editors and Affiliations

  1. Department of Computer Science, Clear Water Bay, Hong Kong University of Science and Technology, Hong Kong, China

    Siu-Wing Cheng

  2. Department of Computer Science, City University of Hong Kong,  

    Chung Keung Poon

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© 2006 Springer-Verlag Berlin Heidelberg

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Taillon, P.J. (2006). A New Approach for Solving the Maximum Clique Problem. In: Cheng, SW., Poon, C.K. (eds) Algorithmic Aspects in Information and Management. AAIM 2006. Lecture Notes in Computer Science, vol 4041. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11775096_26

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  • DOI: https://doi.org/10.1007/11775096_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35157-3

  • Online ISBN: 978-3-540-35158-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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