IDEAS home Printed from https://ideas.repec.org/a/inm/orijoc/v30y2018i1p137-153.html
   My bibliography  Save this article

Incremental Upper Bound for the Maximum Clique Problem

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://doi.org/10.1287/ijoc.2017.0770
    Download Restriction: no
    File URL: https://libkey.io/10.1287/ijoc.2017.0770?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. Jann Michael Weinand & Kenneth Sorensen & Pablo San Segundo & Max Kleinebrahm & Russell McKenna, 2020. "Research trends in combinatorial optimisation," Papers 2012.01294, arXiv.org.
    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. 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. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Laurent, Monique & Vargas, Luis Felipe, 2022. "Finite convergence of sum-of-squares hierarchies for the stability number of a graph," Other publications TiSEM 3998b864-7504-4cf4-bc1d-f, Tilburg University, School of Economics and Management.
    2. Foad Mahdavi Pajouh, 2020. "Minimum cost edge blocker clique problem," Annals of Operations Research, Springer, vol. 294(1), pages 345-376, November.
    3. Zhou, Yi & Lin, Weibo & Hao, Jin-Kao & Xiao, Mingyu & Jin, Yan, 2022. "An effective branch-and-bound algorithm for the maximum s-bundle problem," European Journal of Operational Research, Elsevier, vol. 297(1), pages 27-39.
    4. Filipa D. Carvalho & Maria Teresa Almeida, 2017. "The triangle k-club problem," Journal of Combinatorial Optimization, Springer, vol. 33(3), pages 814-846, April.
    5. Luzhi Wang & Shuli Hu & Mingyang Li & Junping Zhou, 2019. "An Exact Algorithm for Minimum Vertex Cover Problem," Mathematics, MDPI, vol. 7(7), pages 1-8, July.
    6. Zhou, Yi & Rossi, André & Hao, Jin-Kao, 2018. "Towards effective exact methods for the Maximum Balanced Biclique Problem in bipartite graphs," European Journal of Operational Research, Elsevier, vol. 269(3), pages 834-843.
    7. Zhong, Haonan & Mahdavi Pajouh, Foad & A． Prokopyev, Oleg, 2023. "On designing networks resilient to clique blockers," European Journal of Operational Research, Elsevier, vol. 307(1), pages 20-32.
    8. Piotr Łukasiak & Jacek Błażewicz & Maciej Miłostan, 2010. "Some operations research methods for analyzing protein sequences and structures," Annals of Operations Research, Springer, vol. 175(1), pages 9-35, March.
    9. Wang, Yang & Wu, Qinghua & Glover, Fred, 2017. "Effective metaheuristic algorithms for the minimum differential dispersion problem," European Journal of Operational Research, Elsevier, vol. 258(3), pages 829-843.
    10. Rota Bulò, Samuel & Pelillo, Marcello, 2017. "Dominant-set clustering: A review," European Journal of Operational Research, Elsevier, vol. 262(1), pages 1-13.
    11. 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.
    12. Constantine N. Goulimis, 2007. "ASP, The Art and Science of Practice: Appeal to NP-Completeness Considered Harmful: Does the Fact That a Problem Is NP-Complete Tell Us Anything?," Interfaces, INFORMS, vol. 37(6), pages 584-586, December.
    13. Paweł Daniluk & Grzegorz Firlik & Bogdan Lesyng, 2019. "Implementation of a maximum clique search procedure on CUDA," Journal of Heuristics, Springer, vol. 25(2), pages 247-271, April.
    14. Sebastian Lamm & Peter Sanders & Christian Schulz & Darren Strash & Renato F. Werneck, 2017. "Finding near-optimal independent sets at scale," Journal of Heuristics, Springer, vol. 23(4), pages 207-229, August.
    15. James T. Hungerford & Francesco Rinaldi, 2019. "A General Regularized Continuous Formulation for the Maximum Clique Problem," Management Science, INFORMS, vol. 44(4), pages 1161-1173, November.
    16. 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.
    17. Melisew Tefera Belachew & Nicolas Gillis, 2017. "Solving the Maximum Clique Problem with Symmetric Rank-One Non-negative Matrix Approximation," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 279-296, April.
    18. Kristjan Reba & Matej Guid & Kati Rozman & Dušanka Janežič & Janez Konc, 2021. "Exact Maximum Clique Algorithm for Different Graph Types Using Machine Learning," Mathematics, MDPI, vol. 10(1), pages 1-14, December.
    19. Elisabeth Gaar & Melanie Siebenhofer & Angelika Wiegele, 2022. "An SDP-based approach for computing the stability number of a graph," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 95(1), pages 141-161, February.
    20. Douglas Altner & Özlem Ergun, 2011. "Rapidly computing robust minimum capacity s-t cuts: a case study in solving a sequence of maximum flow problems," Annals of Operations Research, Springer, vol. 184(1), pages 3-26, April.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:orijoc:v:30:y:2018:i:1:p:137-153. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.