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

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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.

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Acknowledgements

We would like to thank two anonymous referees, whose feedback helped us to improve the paper. This work was carried out while the 2nd author was a visiting scholar at Texas A&M University, College Station, TX, USA and is partially supported by scholarship SFRH/BSAB/113662/2015. Partial support by DOD-ONR (N00014-13-1-0635) and NSF (CMMI-1538493) Grants is also gratefully acknowledged.

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Authors and Affiliations

  1. Texas A&M University, College Station, TX, USA

    Seyedmohammadhossein Hosseinian & Sergiy Butenko

  2. INESC TEC, Universidade do Porto, Rua Dr. Roberto Frias, 4200-464, Porto, Portugal

    Dalila B. M. M. Fontes

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  1. Seyedmohammadhossein Hosseinian
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  3. Sergiy Butenko
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Correspondence to Sergiy Butenko.

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Hosseinian, S., Fontes, D.B. & Butenko, S. A nonconvex quadratic optimization approach to the maximum edge weight clique problem. J Glob Optim 72, 219–240 (2018). https://doi.org/10.1007/s10898-018-0630-5

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