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Incremental Upper Bound for the Maximum Clique Problem

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  • Chu-Min Li

    (School of Computer Science, Huazhong University of Science and Technology (HUST), Wuhan 430073, China; Modélisation, Information and Systèmes (MIS), EA 4290, Université de Picardie Jules Verne, Amiens 80000, France)

  • Zhiwen Fang

    (Modélisation, Information and Systèmes (MIS), EA 4290, Université de Picardie Jules Verne, Amiens 80000, France; State Key Laboratory of Software Development Environment, Beihang University, Beijing 100191, China)

  • Hua Jiang

    (School of Computer Science, Huazhong University of Science and Technology (HUST), Wuhan 430073, China)

  • Ke Xu

    (State Key Laboratory of Software Development Environment, Beihang University, Beijing 100191, China)

Abstract

The maximum clique problem (MaxClique for short) consists of searching for a maximum complete subgraph in a graph. A branch-and-bound (BnB) MaxClique algorithm computes an upper bound of the number of vertices of a maximum clique at every search tree node, to prune the subtree rooted at the node. Existing upper bounds are usually computed from scratch at every search tree node. In this paper, we define an incremental upper bound, called IncUB, which is derived efficiently from previous searches instead of from scratch. Then, we describe a new BnB MaxClique algorithm, called IncMC2, which uses graph coloring and MaxSAT reasoning to filter out the vertices that do not need to be branched on, and uses IncUB to prune the remaining branches. Our experimental results show that IncMC2 is significantly faster than algorithms such as BBMC and IncMaxCLQ. Finally, we carry out experiments to provide evidence that the performance of IncMC2 is due to IncUB.

Suggested Citation

  • 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.
    • Handle: RePEc:inm:orijoc:v:30:y:2018:i:1:p:137-153
    DOI: 10.1287/ijoc.2017.0770
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    References listed on IDEAS

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    1. Wu, Qinghua & Hao, Jin-Kao, 2015. "A review on algorithms for maximum clique problems," European Journal of Operational Research, Elsevier, vol. 242(3), pages 693-709.
    2. Dawn M. Strickland & Earl Barnes & Joel S. Sokol, 2005. "Optimal Protein Structure Alignment Using Maximum Cliques," Operations Research, INFORMS, vol. 53(3), pages 389-402, June.
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    Cited by:

    1. Jann Michael Weinand & Kenneth Sorensen & Pablo San Segundo & Max Kleinebrahm & Russell McKenna, 2020. "Research trends in combinatorial optimisation," Papers 2012.01294, arXiv.org.
    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. 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.
    4. 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.
    5. 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.
    6. 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.

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