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An exact algorithm for the maximum probabilistic clique problem

Author

Listed:
  • Zhuqi Miao

    (Oklahoma State University)

  • Balabhaskar Balasundaram

    (Oklahoma State University)

  • Eduardo L. Pasiliao

    (Munitions Directorate, Air Force Research Laboratory)

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 $$\theta \in [0,1]$$ θ ∈ [ 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.

Suggested Citation

  • Zhuqi Miao & Balabhaskar Balasundaram & Eduardo L. Pasiliao, 2014. "An exact algorithm for the maximum probabilistic clique problem," Journal of Combinatorial Optimization, Springer, vol. 28(1), pages 105-120, July.
    • Handle: RePEc:spr:jcomop:v:28:y:2014:i:1:d:10.1007_s10878-013-9699-4
    DOI: 10.1007/s10878-013-9699-4
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    References listed on IDEAS

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    1. Svyatoslav Trukhanov & Chitra Balasubramaniam & Balabhaskar Balasundaram & Sergiy Butenko, 2013. "Algorithms for detecting optimal hereditary structures in graphs, with application to clique relaxations," Computational Optimization and Applications, Springer, vol. 56(1), pages 113-130, September.
    2. B. K. Pagnoncelli & S. Ahmed & A. Shapiro, 2009. "Sample Average Approximation Method for Chance Constrained Programming: Theory and Applications," Journal of Optimization Theory and Applications, Springer, vol. 142(2), pages 399-416, August.
    3. E. C. Sewell, 1998. "A Branch and Bound Algorithm for the Stability Number of a Sparse Graph," INFORMS Journal on Computing, INFORMS, vol. 10(4), pages 438-447, November.
    4. R. Luce & Albert Perry, 1949. "A method of matrix analysis of group structure," Psychometrika, Springer;The Psychometric Society, vol. 14(2), pages 95-116, June.
    5. Dorit S. Hochbaum & David B. Shmoys, 1985. "A Best Possible Heuristic for the k -Center Problem," Mathematics of Operations Research, INFORMS, vol. 10(2), pages 180-184, May.
    6. Butenko, S. & Wilhelm, W.E., 2006. "Clique-detection models in computational biochemistry and genomics," European Journal of Operational Research, Elsevier, vol. 173(1), pages 1-17, August.
    7. Jeffrey Pattillo & Nataly Youssef & Sergiy Butenko, 2012. "Clique Relaxation Models in Social Network Analysis," Springer Optimization and Its Applications, in: My T. Thai & Panos M. Pardalos (ed.), Handbook of Optimization in Complex Networks, chapter 0, pages 143-162, Springer.
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    Cited by:

    1. Oleksandra Yezerska & Sergiy Butenko & Vladimir L. Boginski, 2018. "Detecting robust cliques in graphs subject to uncertain edge failures," Annals of Operations Research, Springer, vol. 262(1), pages 109-132, March.

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