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Algorithms for the maximum k-club problem in graphs

Author

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  • Shahram Shahinpour

    (Texas A&M University)

  • Sergiy Butenko

    (Texas A&M University)

Abstract

Given a simple undirected graph G, a k-club is a subset of vertices inducing a subgraph of diameter at most k. The maximum k-club problem (MkCP) is to find a k-club of maximum cardinality in G. These structures, originally introduced to model cohesive subgroups in social network analysis, are of interest in network-based data mining and clustering applications. The maximum k-club problem is NP-hard, moreover, determining whether a given k-club is maximal (by inclusion) is NP-hard as well. This paper first provides a sufficient condition for testing maximality of a given k-club. Then it proceeds to develop a variable neighborhood search (VNS) heuristic and an exact algorithm for MkCP that uses the VNS solution as a lower bound. Computational experiments with test instances available in the literature show that the proposed algorithms are very effective on sparse instances and outperform the existing methods on most dense graphs from the testbed.

Suggested Citation

  • Shahram Shahinpour & Sergiy Butenko, 2013. "Algorithms for the maximum k-club problem in graphs," Journal of Combinatorial Optimization, Springer, vol. 26(3), pages 520-554, October.
    • Handle: RePEc:spr:jcomop:v:26:y:2013:i:3:d:10.1007_s10878-012-9473-z
    DOI: 10.1007/s10878-012-9473-z
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    References listed on IDEAS

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    1. Carvalho, Filipa D. & Almeida, M. Teresa, 2011. "Upper bounds and heuristics for the 2-club problem," European Journal of Operational Research, Elsevier, vol. 210(3), pages 489-494, May.
    2. Veremyev, Alexander & Boginski, Vladimir, 2012. "Identifying large robust network clusters via new compact formulations of maximum k-club problems," European Journal of Operational Research, Elsevier, vol. 218(2), pages 316-326.
    3. 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.
    4. Bourjolly, Jean-Marie & Laporte, Gilbert & Pesant, Gilles, 2002. "An exact algorithm for the maximum k-club problem in an undirected graph," European Journal of Operational Research, Elsevier, vol. 138(1), pages 21-28, April.
    5. Robert Mokken, 1979. "Cliques, clubs and clans," Quality & Quantity: International Journal of Methodology, Springer, vol. 13(2), pages 161-173, April.
    6. Avanthay, Cedric & Hertz, Alain & Zufferey, Nicolas, 2003. "A variable neighborhood search for graph coloring," European Journal of Operational Research, Elsevier, vol. 151(2), pages 379-388, December.
    7. Hansen, Pierre & Mladenovic, Nenad, 2001. "Variable neighborhood search: Principles and applications," European Journal of Operational Research, Elsevier, vol. 130(3), pages 449-467, May.
    8. Gavranovic, Haris, 2008. "Local search and suffix tree for car-sequencing problem with colors," European Journal of Operational Research, Elsevier, vol. 191(3), pages 972-980, December.
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    Cited by:

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    3. Yezerska, Oleksandra & Mahdavi Pajouh, Foad & Butenko, Sergiy, 2017. "On biconnected and fragile subgraphs of low diameter," European Journal of Operational Research, Elsevier, vol. 263(2), pages 390-400.
    4. Timo Gschwind & Stefan Irnich & Fabio Furini & Roberto Wolfler Calvo, 2017. "A Branch-and-Price Framework for Decomposing Graphs into Relaxed Cliques," Working Papers 1723, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.
    5. Timo Gschwind & Stefan Irnich & Fabio Furini & Roberto Wolfler Calvo, 2021. "A Branch-and-Price Framework for Decomposing Graphs into Relaxed Cliques," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 1070-1090, July.
    6. Yuichi Asahiro & Tomohiro Kubo & Eiji Miyano, 2019. "Experimental Evaluation of Approximation and Heuristic Algorithms for Maximum Distance-Bounded Subgraph Problems," The Review of Socionetwork Strategies, Springer, vol. 13(2), pages 143-161, October.
    7. Timo Gschwind & Stefan Irnich & Fabio Furini & Roberto Wol?er Calvo, 2015. "Social Network Analysis and Community Detection by Decomposing a Graph into Relaxed Cliques," Working Papers 1520, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.

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