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An analytical comparison of the LP relaxations of integer models for the k-club problem

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  • Almeida, Maria Teresa
  • Carvalho, Filipa D.

Abstract

Given an undirected graph G=(V,E), a k-club is a subset of nodes that induces a subgraph with diameter at most k. The k-club problem is to find a maximum cardinality k-club. In this study, we use a linear programming relaxation standpoint to compare integer formulations for the k-club problem. The comparisons involve formulations known from the literature and new formulations, built in different variable spaces. For the case k=3, we propose two enhanced compact formulations. From the LP relaxation standpoint these formulations dominate all other compact formulations in the literature and are equivalent to a formulation with a non-polynomial number of constraints. Also for k=3, we compare the relative strength of LP relaxations for all formulations examined in the study (new and known from the literature). Based on insights obtained from the comparative study, we devise a strengthened version of a recursive compact formulation in the literature for the k-club problem (k>1) and show how to modify one of the new formulations for the case k=3 in order to accommodate additional constraints recently proposed in the literature.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:ejores:v:232:y:2014:i:3:p:489-498
    DOI: 10.1016/j.ejor.2013.08.004
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    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. 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.
    3. 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.
    4. Zhou, Yi & Lin, Weibo & Hao, Jin-Kao & Xiao, Mingyu & Jin, Yan, 2022. "An effective branch-and-bound algorithm for the maximum s-bundle problem," European Journal of Operational Research, Elsevier, vol. 297(1), pages 27-39.
    5. Yajun Lu & Hosseinali Salemi & Balabhaskar Balasundaram & Austin Buchanan, 2022. "On Fault-Tolerant Low-Diameter Clusters in Graphs," INFORMS Journal on Computing, INFORMS, vol. 34(6), pages 3181-3199, November.

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