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A dynamic programming algorithm for the maximum s-club problem on trees

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  • Fernández-Zepeda, José Alberto
  • Flores-Lamas, Alejandro
  • Hague, Matthew
  • Trejo-Sánchez, Joel Antonio

Abstract

Computing cliques in an undirected graph G=(VG,EG) is a fundamental problem in social network analysis. However, in some cases, the strict definition of a clique (a subset of vertices pairwise adjacent in G) often limits its applicability in real-world settings. To address this issue, we study the s-club: a clique relaxation that induces a subgraph of diameter at most s. Note that a clique is simply a 1-club. Computing a maximum s-club is a computationally challenging problem, as it is NP-hard for any positive integer s in arbitrary graphs. Thus, this paper presents a simple dynamic programming algorithm that efficiently computes a maximum s-club on an n-vertex tree in O(s⋅n) time. This algorithm outperforms existing algorithms for trees in theory and practice. This approach is a stepping stone towards computing maximum s-clubs on tree-like graphs.

Suggested Citation

  • Fernández-Zepeda, José Alberto & Flores-Lamas, Alejandro & Hague, Matthew & Trejo-Sánchez, Joel Antonio, 2026. "A dynamic programming algorithm for the maximum s-club problem on trees," European Journal of Operational Research, Elsevier, vol. 329(2), pages 426-435.
    • Handle: RePEc:eee:ejores:v:329:y:2026:i:2:p:426-435
    DOI: 10.1016/j.ejor.2025.08.031
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    References listed on IDEAS

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    1. 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.
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    11. Oleksandra Yezerska & Foad Mahdavi Pajouh & Alexander Veremyev & Sergiy Butenko, 2019. "Exact algorithms for the minimum s-club partitioning problem," Annals of Operations Research, Springer, vol. 276(1), pages 267-291, May.
    12. 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.
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