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Efficient branch-and-bound algorithms for finding triangle-constrained 2-clubs

Author

Listed:
  • Niels Grüttemeier

    (Fraunhofer IOSB-INA)

  • Philipp Heinrich Keßler

    (Philipps-Universität Marburg)

  • Christian Komusiewicz

    (Friedrich-Schiller-Universität Jena)

  • Frank Sommer

    (TU Wien)

Abstract

In the Vertex Triangle 2-Club problem, we are given an undirected graph G and aim to find a maximum-vertex subgraph of G that has diameter at most 2 and in which every vertex is contained in at least $$\ell $$ ℓ triangles in the subgraph. So far, the only algorithm for solving Vertex Triangle 2-Club relies on an ILP formulation (Almeida and Brás in Comput Oper Res 111:258–270, 2019). In this work, we develop a combinatorial branch-and-bound algorithm that, coupled with a set of data reduction rules, outperforms the existing implementation and is able to find optimal solutions on sparse real-world graphs with more than 100,000 vertices in a few minutes. We also extend our algorithm to the Edge Triangle 2-Club problem where the triangle constraint is imposed on all edges of the subgraph.

Suggested Citation

  • Niels Grüttemeier & Philipp Heinrich Keßler & Christian Komusiewicz & Frank Sommer, 2024. "Efficient branch-and-bound algorithms for finding triangle-constrained 2-clubs," Journal of Combinatorial Optimization, Springer, vol. 48(3), pages 1-27, October.
    • Handle: RePEc:spr:jcomop:v:48:y:2024:i:3:d:10.1007_s10878-024-01204-z
    DOI: 10.1007/s10878-024-01204-z
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    References listed on IDEAS

    as
    1. 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.
    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. 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.
    4. 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.
    5. 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.
    6. Robert Mokken, 1979. "Cliques, clubs and clans," Quality & Quantity: International Journal of Methodology, Springer, vol. 13(2), pages 161-173, April.
    7. Foad Mahdavi Pajouh & Esmaeel Moradi & Balabhaskar Balasundaram, 2017. "Detecting large risk-averse 2-clubs in graphs with random edge failures," Annals of Operations Research, Springer, vol. 249(1), pages 55-73, February.
    8. 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.
    9. Pattillo, Jeffrey & Youssef, Nataly & Butenko, Sergiy, 2013. "On clique relaxation models in network analysis," European Journal of Operational Research, Elsevier, vol. 226(1), pages 9-18.
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