IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i18p2850-d1477804.html
   My bibliography  Save this article

Probabilistic Cellular Automata Monte Carlo for the Maximum Clique Problem

Author

Listed:
  • Alessio Troiani

    (Dipartimento di Matematica e Informatica, Università di Perugia, Via Vanvitelli, 06123 Perugia, Italy)

Abstract

We consider the problem of finding the largest clique of a graph. This is an NP-hard problem and no exact algorithm to solve it exactly in polynomial time is known to exist. Several heuristic approaches have been proposed to find approximate solutions. Markov Chain Monte Carlo is one of these. In the context of Markov Chain Monte Carlo, we present a class of “parallel dynamics”, known as Probabilistic Cellular Automata, which can be used in place of the more standard choice of sequential “single spin flip” to sample from a probability distribution concentrated on the largest cliques of the graph. We perform a numerical comparison between the two classes of chains both in terms of the quality of the solution and in terms of computational time. We show that the parallel dynamics are considerably faster than the sequential ones while providing solutions of comparable quality.

Suggested Citation

  • Alessio Troiani, 2024. "Probabilistic Cellular Automata Monte Carlo for the Maximum Clique Problem," Mathematics, MDPI, vol. 12(18), pages 1-16, September.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:18:p:2850-:d:1477804
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/18/2850/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/18/2850/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Juho Lauri & Sourav Dutta & Marco Grassia & Deepak Ajwani, 2023. "Learning fine-grained search space pruning and heuristics for combinatorial optimization," Journal of Heuristics, Springer, vol. 29(2), pages 313-347, June.
    2. 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.
    3. Fred Glover & Gary Kochenberger & Yu Du, 2019. "Quantum Bridge Analytics I: a tutorial on formulating and using QUBO models," 4OR, Springer, vol. 17(4), pages 335-371, December.
    4. Fred Glover & Gary Kochenberger & Moses Ma & Yu Du, 2022. "Quantum Bridge Analytics II: QUBO-Plus, network optimization and combinatorial chaining for asset exchange," Annals of Operations Research, Springer, vol. 314(1), pages 185-212, July.
    5. R. Luce & Albert Perry, 1949. "A method of matrix analysis of group structure," Psychometrika, Springer;The Psychometric Society, vol. 14(2), pages 95-116, June.
    Full references (including those not matched with items on IDEAS)

    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. Oleksandra Yezerska & Sergiy Butenko & Vladimir L. Boginski, 2018. "Detecting robust cliques in graphs subject to uncertain edge failures," Annals of Operations Research, Springer, vol. 262(1), pages 109-132, March.
    2. 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.
    3. Immanuel. M. Bomze & Francesco Rinaldi & Damiano Zeffiro, 2024. "Frank–Wolfe and friends: a journey into projection-free first-order optimization methods," Annals of Operations Research, Springer, vol. 343(2), pages 607-638, December.
    4. Immanuel M. Bomze & Francesco Rinaldi & Damiano Zeffiro, 2021. "Frank–Wolfe and friends: a journey into projection-free first-order optimization methods," 4OR, Springer, vol. 19(3), pages 313-345, September.
    5. Seyedmohammadhossein Hosseinian & Dalila B. M. M. Fontes & Sergiy Butenko, 2020. "A Lagrangian Bound on the Clique Number and an Exact Algorithm for the Maximum Edge Weight Clique Problem," INFORMS Journal on Computing, INFORMS, vol. 32(3), pages 747-762, July.
    6. Yves Crama & Michel Grabisch & Silvano Martello, 2022. "Preface," Annals of Operations Research, Springer, vol. 314(1), pages 1-3, July.
    7. Byron Tasseff & Tameem Albash & Zachary Morrell & Marc Vuffray & Andrey Y. Lokhov & Sidhant Misra & Carleton Coffrin, 2024. "On the emerging potential of quantum annealing hardware for combinatorial optimization," Journal of Heuristics, Springer, vol. 30(5), pages 325-358, December.
    8. Simone Celant, 2013. "Two-mode networks: the measurement of efficiency in the profiles of actors’ participation in the occasions," Quality & Quantity: International Journal of Methodology, Springer, vol. 47(6), pages 3289-3302, October.
    9. Noah E. Friedkin, 1984. "Structural Cohesion and Equivalence Explanations of Social Homogeneity," Sociological Methods & Research, , vol. 12(3), pages 235-261, February.
    10. 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.
    11. Le Breton, Michel & Weber, Shlomo, 2009. "Existence of Pure Strategies Nash Equilibria in Social Interaction Games with Dyadic Externalities," CEPR Discussion Papers 7279, C.E.P.R. Discussion Papers.
    12. Lehouillier, Thibault & Omer, Jérémy & Soumis, François & Desaulniers, Guy, 2017. "Two decomposition algorithms for solving a minimum weight maximum clique model for the air conflict resolution problem," European Journal of Operational Research, Elsevier, vol. 256(3), pages 696-712.
    13. Foad Mahdavi Pajouh, 2020. "Minimum cost edge blocker clique problem," Annals of Operations Research, Springer, vol. 294(1), pages 345-376, November.
    14. Zhu, Yongjun & Yan, Erjia, 2017. "Examining academic ranking and inequality in library and information science through faculty hiring networks," Journal of Informetrics, Elsevier, vol. 11(2), pages 641-654.
    15. Zhuqi Miao & Balabhaskar Balasundaram & Eduardo L. Pasiliao, 2014. "An exact algorithm for the maximum probabilistic clique problem," Journal of Combinatorial Optimization, Springer, vol. 28(1), pages 105-120, July.
    16. Eric van Diessen & Willemiek J E M Zweiphenning & Floor E Jansen & Cornelis J Stam & Kees P J Braun & Willem M Otte, 2014. "Brain Network Organization in Focal Epilepsy: A Systematic Review and Meta-Analysis," PLOS ONE, Public Library of Science, vol. 9(12), pages 1-21, December.
    17. Henderson, Geraldine R. & Iacobucci, Dawn & Calder, Bobby J., 1998. "Brand diagnostics: Mapping branding effects using consumer associative networks," European Journal of Operational Research, Elsevier, vol. 111(2), pages 306-327, December.
    18. 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.
    19. Etienne Farvaque & Frédéric Gannon, 2018. "Profiling giants: the networks and influence of Buchanan and Tullock," Public Choice, Springer, vol. 175(3), pages 277-302, June.
    20. Zhou, Yi & Hao, Jin-Kao & Goëffon, Adrien, 2017. "PUSH: A generalized operator for the Maximum Vertex Weight Clique Problem," European Journal of Operational Research, Elsevier, vol. 257(1), pages 41-54.

    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:gam:jmathe:v:12:y:2024:i:18:p:2850-:d:1477804. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.