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A nonconvex quadratic optimization approach to the maximum edge weight clique problem

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  • Seyedmohammadhossein Hosseinian

    (Texas A&M University)

  • Dalila B. M. M. Fontes

    (Universidade do Porto)

  • Sergiy Butenko

    (Texas A&M University)

Abstract

The maximum edge weight clique (MEWC) problem, defined on a simple edge-weighted graph, is to find a subset of vertices inducing a complete subgraph with the maximum total sum of edge weights. We propose a quadratic optimization formulation for the MEWC problem and study characteristics of this formulation in terms of local and global optimality. We establish the correspondence between local maxima of the proposed formulation and maximal cliques of the underlying graph, implying that the characteristic vector of a MEWC in the graph is a global optimizer of the continuous problem. In addition, we present an exact algorithm to solve the MEWC problem. The algorithm is a combinatorial branch-and-bound procedure that takes advantage of a new upper bound as well as an efficient construction heuristic based on the proposed quadratic formulation. Results of computational experiments on some benchmark instances are also presented.

Suggested Citation

  • Seyedmohammadhossein Hosseinian & Dalila B. M. M. Fontes & Sergiy Butenko, 2018. "A nonconvex quadratic optimization approach to the maximum edge weight clique problem," Journal of Global Optimization, Springer, vol. 72(2), pages 219-240, October.
    • Handle: RePEc:spr:jglopt:v:72:y:2018:i:2:d:10.1007_s10898-018-0630-5
    DOI: 10.1007/s10898-018-0630-5
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    References listed on IDEAS

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    1. Seyedmohammadhossein Hosseinian & Dalila B. M. M. Fontes & Sergiy Butenko & Marco Buongiorno Nardelli & Marco Fornari & Stefano Curtarolo, 2017. "The Maximum Edge Weight Clique Problem: Formulations and Solution Approaches," Springer Optimization and Its Applications, in: Sergiy Butenko & Panos M. Pardalos & Volodymyr Shylo (ed.), Optimization Methods and Applications, pages 217-237, Springer.
    2. Macambira, Elder Magalhaes & de Souza, Cid Carvalho, 2000. "The edge-weighted clique problem: Valid inequalities, facets and polyhedral computations," European Journal of Operational Research, Elsevier, vol. 123(2), pages 346-371, June.
    3. Alidaee, Bahram & Glover, Fred & Kochenberger, Gary & Wang, Haibo, 2007. "Solving the maximum edge weight clique problem via unconstrained quadratic programming," European Journal of Operational Research, Elsevier, vol. 181(2), pages 592-597, September.
    4. 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.
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    9. Stanislav Busygin & Sergiy Butenko & Panos M. Pardalos, 2002. "A Heuristic for the Maximum Independent Set Problem Based on Optimization of a Quadratic Over a Sphere," Journal of Combinatorial Optimization, Springer, vol. 6(3), pages 287-297, September.
    10. Mikhail Batsyn & Boris Goldengorin & Evgeny Maslov & Panos M. Pardalos, 2014. "Improvements to MCS algorithm for the maximum clique problem," Journal of Combinatorial Optimization, Springer, vol. 27(2), pages 397-416, February.
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    1. 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.
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    3. Stefano Coniglio & Stefano Gualandi, 2022. "Optimizing over the Closure of Rank Inequalities with a Small Right-Hand Side for the Maximum Stable Set Problem via Bilevel Programming," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 1006-1023, March.
    4. 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.

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