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A branch-and-cut algorithm for the Edge Interdiction Clique Problem

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  • Furini, Fabio
  • Ljubić, Ivana
  • San Segundo, Pablo
  • Zhao, Yanlu

Abstract

Given a graph G and an interdiction budget k∈N, the Edge Interdiction Clique Problem (EICP) asks to find a subset of at most k edges to remove from G so that the size of the maximum clique, in the interdicted graph, is minimized. The EICP belongs to the family of interdiction problems with the aim of reducing the clique number of the graph. The EICP optimal solutions, called optimal interdiction policies, determine the subset of most vital edges of a graph which are crucial for preserving its clique number. We propose a new set-covering-based Integer Linear Programming (ILP) formulation for the EICP with an exponential number of constraints, called the clique-covering inequalities. We design a new branch-and-cut algorithm which is enhanced by a tailored separation procedure and by an effective heuristic initialization phase. Thanks to the new exact algorithm, we manage to solve the EICP in several sets of instances from the literature. Extensive tests show that the new exact algorithm greatly outperforms the state-of-the-art approaches for the EICP.

Suggested Citation

  • Furini, Fabio & Ljubić, Ivana & San Segundo, Pablo & Zhao, Yanlu, 2021. "A branch-and-cut algorithm for the Edge Interdiction Clique Problem," European Journal of Operational Research, Elsevier, vol. 294(1), pages 54-69.
    • Handle: RePEc:eee:ejores:v:294:y:2021:i:1:p:54-69
    DOI: 10.1016/j.ejor.2021.01.030
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    References listed on IDEAS

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    1. Furini, Fabio & Ljubić, Ivana & Martin, Sébastien & San Segundo, Pablo, 2019. "The maximum clique interdiction problem," European Journal of Operational Research, Elsevier, vol. 277(1), pages 112-127.
    2. Matteo Fischetti & Ivana Ljubić & Michele Monaci & Markus Sinnl, 2017. "A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs," Operations Research, INFORMS, vol. 65(6), pages 1615-1637, December.
    3. Li, Chu-Min & Liu, Yanli & Jiang, Hua & Manyà, Felip & Li, Yu, 2018. "A new upper bound for the maximum weight clique problem," European Journal of Operational Research, Elsevier, vol. 270(1), pages 66-77.
    4. Coniglio, Stefano & Furini, Fabio & San Segundo, Pablo, 2021. "A new combinatorial branch-and-bound algorithm for the Knapsack Problem with Conflicts," European Journal of Operational Research, Elsevier, vol. 289(2), pages 435-455.
    5. Chu-Min Li & Zhiwen Fang & Hua Jiang & Ke Xu, 2018. "Incremental Upper Bound for the Maximum Clique Problem," INFORMS Journal on Computing, INFORMS, vol. 30(1), pages 137-153, February.
    6. San Segundo, Pablo & Coniglio, Stefano & Furini, Fabio & Ljubić, Ivana, 2019. "A new branch-and-bound algorithm for the maximum edge-weighted clique problem," European Journal of Operational Research, Elsevier, vol. 278(1), pages 76-90.
    7. Qinghua Wu & Jin-Kao Hao, 2013. "An adaptive multistart tabu search approach to solve the maximum clique problem," Journal of Combinatorial Optimization, Springer, vol. 26(1), pages 86-108, July.
    8. Fischetti, Matteo & Monaci, Michele & Sinnl, Markus, 2018. "A dynamic reformulation heuristic for Generalized Interdiction Problems," European Journal of Operational Research, Elsevier, vol. 267(1), pages 40-51.
    9. Álvarez-Miranda, Eduardo & Goycoolea, Marcos & Ljubić, Ivana & Sinnl, Markus, 2021. "The Generalized Reserve Set Covering Problem with Connectivity and Buffer Requirements," European Journal of Operational Research, Elsevier, vol. 289(3), pages 1013-1029.
    10. Mohammed-Albarra Hassan & Imed Kacem & Sébastien Martin & Izzeldin M. Osman, 2018. "On the m-clique free interval subgraphs polytope: polyhedral analysis and applications," Journal of Combinatorial Optimization, Springer, vol. 36(3), pages 1074-1101, October.
    11. Fabio Furini & Manuel Iori & Silvano Martello & Mutsunori Yagiura, 2015. "Heuristic and Exact Algorithms for the Interval Min–Max Regret Knapsack Problem," INFORMS Journal on Computing, INFORMS, vol. 27(2), pages 392-405, May.
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    Cited by:

    1. Cerulli, Martina & Serra, Domenico & Sorgente, Carmine & Archetti, Claudia & Ljubić, Ivana, 2023. "Mathematical programming formulations for the Collapsed k-Core Problem," European Journal of Operational Research, Elsevier, vol. 311(1), pages 56-72.
    2. San Segundo, Pablo & Furini, Fabio & León, Rafael, 2022. "A new branch-and-filter exact algorithm for binary constraint satisfaction problems," European Journal of Operational Research, Elsevier, vol. 299(2), pages 448-467.
    3. Wei, Ningji & Walteros, Jose L., 2022. "Integer programming methods for solving binary interdiction games," European Journal of Operational Research, Elsevier, vol. 302(2), pages 456-469.
    4. Leitner, Markus & Ljubić, Ivana & Monaci, Michele & Sinnl, Markus & Tanınmış, Kübra, 2023. "An exact method for binary fortification games," European Journal of Operational Research, Elsevier, vol. 307(3), pages 1026-1039.
    5. San Segundo, Pablo & Furini, Fabio & Álvarez, David & Pardalos, Panos M., 2023. "CliSAT: A new exact algorithm for hard maximum clique problems," European Journal of Operational Research, Elsevier, vol. 307(3), pages 1008-1025.
    6. Kübra Tanınmış & Markus Sinnl, 2022. "A Branch-and-Cut Algorithm for Submodular Interdiction Games," INFORMS Journal on Computing, INFORMS, vol. 34(5), pages 2634-2657, September.

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