Journal of Combinatorial Optimization

, Volume 28, Issue 1, pp 105–120 | Cite as

An exact algorithm for the maximum probabilistic clique problem

  • Zhuqi Miao
  • Balabhaskar Balasundaram
  • Eduardo L. Pasiliao
Article

Abstract

The maximum clique problem is a classical problem in combinatorial optimization that has a broad range of applications in graph-based data mining, social and biological network analysis and a variety of other fields. This article investigates the problem when the edges fail independently with known probabilities. This leads to the maximum probabilistic clique problem, which is to find a subset of vertices of maximum cardinality that forms a clique with probability at least \(\theta \in [0,1]\), which is a user-specified probability threshold. We show that the probabilistic clique property is hereditary and extend a well-known exact combinatorial algorithm for the maximum clique problem to a sampling-free exact algorithm for the maximum probabilistic clique problem. The performance of the algorithm is benchmarked on a test-bed of DIMACS clique instances and on a randomly generated test-bed.

Keywords

Maximum clique problem Probabilistic programming  Probabilistic clique Branch-and-bound 

References

  1. Ahmed S (2006) Convexity and decomposition of mean-risk stochastic programs. Math Progr 106:433–446CrossRefMATHGoogle Scholar
  2. Ahmed S, Shapiro A (2008) Solving chance-constrained stochastic programs via sampling and integer programming. In: Chen ZL, Raghavan S (eds) Tutorials in operations research, 10th edn. INFORMS, MinneapolisGoogle Scholar
  3. Applegate D, Johnson DS (1988) dfmax.c [C program], available online. ftp://dimacs.rutgers.edu/pub/challenge/graph/solvers/dfmax.c
  4. Balas E, Xue J (1996) Weighted and unweighted maximum clique algorithms with upper bounds from fractional coloring. Algorithmica 15:397–412CrossRefMATHMathSciNetGoogle Scholar
  5. Balas E, Yu C (1986) Finding a maximum clique in an arbitrary graph. SIAM J Comput 15:1054–1068CrossRefMATHMathSciNetGoogle Scholar
  6. Balasundaram B, Butenko S (2008) Network clustering. In: Junker BH, Schreiber F (eds) Analysis of biological networks. Wiley, New York, pp 113–138CrossRefGoogle Scholar
  7. Balasundaram B, Pajouh FM (2013) Graph theoretic clique relaxations and applications. In: Pardalos PM, Du DZ, Graham R (eds) Handbook of combinatorial optimization, 2nd edn. Springer. doi:10.1007/978-1-4419-7997-1_9
  8. Batsyn M, Goldengorin B, Maslov E, Pardalos P (2013) Improvements to mcs algorithm for the maximum clique problem. J Comb Optim 26:1–20. doi:10.1007/s10878-012-9592-6 CrossRefMathSciNetGoogle Scholar
  9. Bertsimas D, Brown DB, Caramanis C (2011) Theory and applications of robust optimization. SIAM Rev 53(3):464–501CrossRefMATHMathSciNetGoogle Scholar
  10. Boginski V (2011) Network-based data mining: operations research techniques and applications. In: Encyclopedia of operations research and management science, Wiley, New YorkGoogle Scholar
  11. Bomze IM, Budinich M, Pardalos PM, Pelillo M (1999) The maximum clique problem. In: Du DZ, Pardalos PM (eds) Handbook of combinatorial optimization. Kluwer Academic, Dordrecht, pp 1–74CrossRefGoogle Scholar
  12. Butenko S, Wilhelm W (2006) Clique-detection models in computational biochemistry and genomics. Eur J Oper Res 173:1–17CrossRefMATHMathSciNetGoogle Scholar
  13. Carraghan R, Pardalos P (1990) An exact algorithm for the maximum clique problem. Oper Res Lett 9:375–382CrossRefMATHGoogle Scholar
  14. Cook DJ, Holder LB (2000) Graph-based data mining. IEEE Intell Syst 15(2):32–41CrossRefGoogle Scholar
  15. DIMACS (1995) Cliques, coloring, and satisfiability: second dimacs implementation challenge. http://dimacs.rutgers.edu/Challenges/
  16. Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. W.H. Freeman and Company, New YorkMATHGoogle Scholar
  17. Harary F, Ross IC (1957) A procedure for clique detection using the group matrix. Sociometry 20:205–215CrossRefMathSciNetGoogle Scholar
  18. Håstad J (1999) Clique is hard to approximate within \(n^{1-\epsilon }\). Acta Math 182:105–142CrossRefMATHMathSciNetGoogle Scholar
  19. Hochbaum DS, Shmoys DB (1985) A best possible heuristic for the \(k\)-center problem. Math Oper Res 10:180–184CrossRefMATHMathSciNetGoogle Scholar
  20. Johnson D, Trick M (eds) (1996) Cliques, coloring, and satisfiablility: second dimacs implementation challenge, DIMACS series in discrete mathematics and theoretical computer science, vol 26. American Mathematical Society, ProvidenceGoogle Scholar
  21. Krokhmal P, Uryasev S, Zrazhevsky G (2005) Numerical comparison of conditional value-at-risk and conditional drawdown-at-risk approaches: application to hedge funds. In: Applications of stochastic programming, MPS/SIAM Ser. Optim., vol 5, SIAM, Philadelphia, pp 609–631Google Scholar
  22. Kubale M (2004) Graph colorings, 352nd edn. American Mathematical Society, ProvidenceCrossRefMATHGoogle Scholar
  23. Luce RD, Perry AD (1949) A method of matrix analysis of group structure. Psychometrika 14(2):95–116CrossRefMathSciNetGoogle Scholar
  24. Luedtke J (2010) An integer programming and decomposition approach to general chance-constrained mathematical programs. In: Eisenbrand F, Shepherd F (eds) Integer programming and combinatorial optimization, lecture notes in computer science, vol 6080. Springer, Berlin / Heidelberg, pp 271–284CrossRefGoogle Scholar
  25. Luedtke J, Ahmed S (2008) A sample approximation approach for optimization with probabilistic constraints. SIAM J Optim 19(2):674–699CrossRefMATHMathSciNetGoogle Scholar
  26. McClosky B (2011) Clique relaxations. In: Encyclopedia of operations research and management science, Wiley, New YorkGoogle Scholar
  27. Nemirovski A, Shapiro A (2004) Scenario approximations of chance constraints. In: Probabilistic and randomized methods for design under uncertainty, Springer, Heidelberg, pp 3–48Google Scholar
  28. Nemirovski A, Shapiro A (2006a) Convex approximations of chance constrained programs. SIAM J Optim 17:969–996CrossRefMATHMathSciNetGoogle Scholar
  29. Nemirovski A, Shapiro A (2006b) Scenario approximations of chance constraints. In: Calafiore G, Dabbene F (eds) Probabilistic and randomized methods for design under uncertainty. Springer, London, pp 3–47CrossRefGoogle Scholar
  30. Östergård PRJ (2002) A fast algorithm for the maximum clique problem. Discrete Appl Math 120:197–207CrossRefMATHMathSciNetGoogle Scholar
  31. Pagnoncelli BK, Ahmed S, Shapiro A (2009) Sample average approximation method for chance constrained programming: theory and applications. J Optim Theory Appl 142:399–416CrossRefMATHMathSciNetGoogle Scholar
  32. Pardalos PM, Xue J (1994) The maximum clique problem. J Glob Optim 4:301–328CrossRefMATHMathSciNetGoogle Scholar
  33. Pattillo J, Youssef N, Butenko S (2012) Clique relaxation models in social network analysis. In: Thai MT, Pardalos PM (eds) Handbook of optimization in complex networks, springer optimization and its applications, vol 58. Springer, New York, pp 143–162CrossRefGoogle Scholar
  34. Prékopa A (2003) Probabilistic programming. In: Ruszczynski A, Shapiro A (eds) Stochastic programming, handbooks in operations research and management, vol 10. Elsevier, Salt Lake, pp 267–351Google Scholar
  35. Rockafellar R, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2(3):21–41Google Scholar
  36. Sewell EC (1998) A branch and bound algorithm for the stability number of a sparse graph. INFORMS J Comput 10(4):438–447CrossRefMathSciNetGoogle Scholar
  37. Shapiro A, Dentcheva D, Ruszczynski A (eds) (2009) Lectures on stochastic programming: modeling and theory. Society for Industrial and Applied Mathematics (SIAM): MPS/SIAM series on optimization, PhiladelphiaGoogle Scholar
  38. Tomita E, Kameda T (2007) An efficient branch-and-bound algorithm for finding a maximum clique with computational experiments. J Glob Optim 37(1):95–111CrossRefMATHMathSciNetGoogle Scholar
  39. Tomita E, Sutani Y, Higashi T, Takahashi S, Wakatsuki M (2010) A simple and faster branch-and-bound algorithm for finding a maximum clique. In: Rahman M, Fujita S (eds) WALCOM: algorithms and computation, lecture notes in computer science, vol 5942. Springer, Berlin Heidelberg, pp 191–203CrossRefGoogle Scholar
  40. Trukhanov S, Balasubramaniam C, Balasundaram B, Butenko S (2013) Algorithms for detecting optimal hereditary structures in graphs, with application to clique relaxations. Comput Optim Appl 56(1):113–130CrossRefMATHMathSciNetGoogle Scholar
  41. Vaskelainen V (2010) Russian doll search algorithms for discrete optimization problems. PhD thesis, Helsinki University of TechnologyGoogle Scholar
  42. Wasserman S, Faust K (1994) Social network analysis. Cambridge University Press, New YorkCrossRefGoogle Scholar
  43. Wood DR (1997) An algorithm for finding a maximum clique in a graph. Oper Res Lett 21(5):211–217CrossRefMATHMathSciNetGoogle Scholar
  44. Yannakakis M (1978) Node-and edge-deletion NP-complete problems. STOC ’78 In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing. ACM Press, New York, pp 253–264Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Zhuqi Miao
    • 1
  • Balabhaskar Balasundaram
    • 1
  • Eduardo L. Pasiliao
    • 2
  1. 1.School of Industrial Engineering & ManagementOklahoma State UniversityStillwater USA
  2. 2.Munitions Directorate, Air Force Research LaboratoryEglin AFBUSA

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