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The triangle k-club problem

Author

Listed:
  • Filipa D. Carvalho

    (ISEG, Universidade de Lisboa
    CMAF-CIO, FC, Universidade de Lisboa
    Instituto Superior de Economia e Gestão)

  • Maria Teresa Almeida

    (ISEG, Universidade de Lisboa
    CMAF-CIO, FC, Universidade de Lisboa)

Abstract

Graph models have long been used in social network analysis and other social and natural sciences to render the analysis of complex systems easier. In applied studies, to understand the behaviour of social networks and the interactions that command that behaviour, it is often necessary to identify sets of elements which form cohesive groups, i.e., groups of actors that are strongly interrelated. The clique concept is a suitable representation for groups of actors that are all directly related pair-wise. However, many social relationships are established not only face-to-face but also through intermediaries, and the clique concept misses all the latter. To deal with these cases, it is necessary to adopt approaches that relax the clique concept. In this paper we introduce a new clique relaxation—the triangle k-club—and its associated maximization problem—the maximum triangle k-club problem. We propose integer programming formulations for the problem, stated in different variable spaces, and derive valid inequalities to strengthen their linear programming relaxations. Computational results on randomly generated and real-world graphs, with $$k=2$$ k = 2 and $$k=3$$ k = 3 , are reported.

Suggested Citation

  • Filipa D. Carvalho & Maria Teresa Almeida, 2017. "The triangle k-club problem," Journal of Combinatorial Optimization, Springer, vol. 33(3), pages 814-846, April.
  • Handle: RePEc:spr:jcomop:v:33:y:2017:i:3:d:10.1007_s10878-016-0009-9
    DOI: 10.1007/s10878-016-0009-9
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    References listed on IDEAS

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    1. Balabhaskar Balasundaram & Sergiy Butenko & Illya V. Hicks, 2011. "Clique Relaxations in Social Network Analysis: The Maximum k -Plex Problem," Operations Research, INFORMS, vol. 59(1), pages 133-142, February.
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    3. 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.
    4. Blazewicz, Jacek & Formanowicz, Piotr & Kasprzak, Marta, 2005. "Selected combinatorial problems of computational biology," European Journal of Operational Research, Elsevier, vol. 161(3), pages 585-597, March.
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    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. Robert Mokken, 1979. "Cliques, clubs and clans," Quality & Quantity: International Journal of Methodology, Springer, vol. 13(2), pages 161-173, April.
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    Cited by:

    1. 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.
    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.

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