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Upper bounds and heuristics for the 2-club problem

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  • Carvalho, Filipa D.
  • Almeida, M. Teresa

Abstract

Given an undirected graph G = (V, E), a k-club is a subset of V that induces a subgraph of diameter at most k. The k-club problem is that of finding the maximum cardinality k-club in G. In this paper we present valid inequalities for the 2-club polytope and derive conditions for them to define facets. These inequalities are the basis of a strengthened formulation for the 2-club problem and a cutting plane algorithm. The LP relaxation of the strengthened formulation is used to compute upper bounds on the problem's optimum and to guide the generation of near-optimal solutions. Numerical experiments indicate that this approach is quite effective in terms of solution quality and speed, especially for low density graphs.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:ejores:v:210:y:2011:i:3:p:489-494
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    References listed on IDEAS

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    Cited by:

    1. Filipa D. Carvalho & Maria Teresa Almeida, 2017. "The triangle k-club problem," Journal of Combinatorial Optimization, Springer, vol. 33(3), pages 814-846, April.
    2. Komusiewicz, Christian & Nichterlein, André & Niedermeier, Rolf & Picker, Marten, 2019. "Exact algorithms for finding well-connected 2-clubs in sparse real-world graphs: Theory and experiments," European Journal of Operational Research, Elsevier, vol. 275(3), pages 846-864.
    3. 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.
    4. Veremyev, Alexander & Boginski, Vladimir & Pasiliao, Eduardo L. & Prokopyev, Oleg A., 2022. "On integer programming models for the maximum 2-club problem and its robust generalizations in sparse graphs," European Journal of Operational Research, Elsevier, vol. 297(1), pages 86-101.
    5. Timo Gschwind & Stefan Irnich & Fabio Furini & Roberto Wolfler Calvo, 2017. "Social Network Analysis and Community Detection by Decomposing a Graph into Relaxed Cliques," Working Papers 1722, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.
    6. 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.
    7. Foad Mahdavi Pajouh & Balabhaskar Balasundaram & Illya V. Hicks, 2016. "On the 2-Club Polytope of Graphs," Operations Research, INFORMS, vol. 64(6), pages 1466-1481, December.
    8. Almeida, Maria Teresa & Carvalho, Filipa D., 2014. "An analytical comparison of the LP relaxations of integer models for the k-club problem," European Journal of Operational Research, Elsevier, vol. 232(3), pages 489-498.

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