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An Ellipsoidal Bounding Scheme for the Quasi-Clique Number of a Graph

Author

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  • Zhuqi Miao

    (Center for Health Systems Innovation, Oklahoma State University, Stillwater, Oklahoma 74078)

  • Balabhaskar Balasundaram

    (School of Industrial Engineering and Management, Oklahoma State University, Stillwater, Oklahoma 74078)

Abstract

A γ -quasi-clique in a simple undirected graph refers to a subset of vertices that induces a subgraph with edge density at least γ. When γ equals one, this definition corresponds to a classical clique. When γ is less than one, it relaxes the requirement of all possible edges by the clique definition. Quasi-clique detection has been used in graph-based data mining to find dense clusters, especially in large-scale error-prone data sets in which the clique model can be overly restrictive. The maximum γ -quasi-clique problem , seeking a γ-quasi-clique of maximum cardinality in the given graph, can be formulated as an optimization problem with a linear objective function and a single quadratic constraint in binary variables. This article investigates the Lagrangian dual of this formulation and develops an upper-bounding technique using the geometry of ellipsoids to bound the Lagrangian dual. The tightness of the upper bound is compared with those obtained from multiple mixed-integer programming formulations of the problem via experiments on benchmark instances.

Suggested Citation

  • Zhuqi Miao & Balabhaskar Balasundaram, 2020. "An Ellipsoidal Bounding Scheme for the Quasi-Clique Number of a Graph," INFORMS Journal on Computing, INFORMS, vol. 32(3), pages 763-778, July.
    • Handle: RePEc:inm:orijoc:v:32:y:3:i:2020:p:763-778
    DOI: 10.1287/ijoc.2019.0922
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    References listed on IDEAS

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    1. Alexander Veremyev & Oleg A. Prokopyev & Sergiy Butenko & Eduardo L. Pasiliao, 2016. "Exact MIP-based approaches for finding maximum quasi-cliques and dense subgraphs," Computational Optimization and Applications, Springer, vol. 64(1), pages 177-214, May.
    2. Foad Mahdavi Pajouh & Zhuqi Miao & Balabhaskar Balasundaram, 2014. "A branch-and-bound approach for maximum quasi-cliques," Annals of Operations Research, Springer, vol. 216(1), pages 145-161, May.
    3. Pinto, Bruno Q. & Ribeiro, Celso C. & Rosseti, Isabel & Plastino, Alexandre, 2018. "A biased random-key genetic algorithm for the maximum quasi-clique problem," European Journal of Operational Research, Elsevier, vol. 271(3), pages 849-865.
    4. 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.
    5. D. Li & X. Sun & C. Liu, 2012. "An exact solution method for unconstrained quadratic 0–1 programming: a geometric approach," Journal of Global Optimization, Springer, vol. 52(4), pages 797-829, April.
    6. 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.
    7. Balabhaskar Balasundaram & Sergiy Butenko & Illya V. Hicks, 2011. "Clique Relaxations in Social Network Analysis: The Maximum k -Plex Problem," Operations Research, INFORMS, vol. 59(1), pages 133-142, February.
    8. 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.
    9. Saban, Daniela & Bonomo, Flavia & Stier-Moses, Nicolás E., 2010. "Analysis and models of bilateral investment treaties using a social networks approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(17), pages 3661-3673.
    10. Balabhaskar Balasundaram & Sergiy Butenko & Svyatoslav Trukhanov, 2005. "Novel Approaches for Analyzing Biological Networks," Journal of Combinatorial Optimization, Springer, vol. 10(1), pages 23-39, August.
    11. Robert Mokken, 1979. "Cliques, clubs and clans," Quality & Quantity: International Journal of Methodology, Springer, vol. 13(2), pages 161-173, April.
    12. R. Luce, 1950. "Connectivity and generalized cliques in sociometric group structure," Psychometrika, Springer;The Psychometric Society, vol. 15(2), pages 169-190, June.
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