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Improving the linear relaxation of maximum k-cut with semidefinite-based constraints

Author

Listed:
  • Vilmar Jefté Rodrigues de Sousa

    (École Polytechnique de Montréal)

  • Miguel F. Anjos

    (École Polytechnique de Montréal)

  • Sébastien Le Digabel

    (École Polytechnique de Montréal)

Abstract

We consider the maximum k-cut problem that involves partitioning the vertex set of a graph into k subsets such that the sum of the weights of the edges joining vertices in different subsets is maximized. The associated semidefinite programming (SDP) relaxation is known to provide strong bounds, but it has a high computational cost. We use a cutting-plane algorithm that relies on the early termination of an interior point method, and we study the performance of SDP and linear programming (LP) relaxations for various values of k and instance types. The LP relaxation is strengthened using combinatorial facet-defining inequalities and SDP-based constraints. Our computational results suggest that the LP approach, especially with the addition of SDP-based constraints, outperforms the SDP relaxations for graphs with positive-weight edges and $$k \ge 7$$ k ≥ 7 .

Suggested Citation

  • Vilmar Jefté Rodrigues de Sousa & Miguel F. Anjos & Sébastien Le Digabel, 2019. "Improving the linear relaxation of maximum k-cut with semidefinite-based constraints," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 7(2), pages 123-151, June.
    • Handle: RePEc:spr:eurjco:v:7:y:2019:i:2:d:10.1007_s13675-019-00110-y
    DOI: 10.1007/s13675-019-00110-y
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    References listed on IDEAS

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    1. Fuda Ma & Jin-Kao Hao, 2017. "A multiple search operator heuristic for the max-k-cut problem," Annals of Operations Research, Springer, vol. 248(1), pages 365-403, January.
    2. Vilmar Jefté Rodrigues de Sousa & Miguel F. Anjos & Sébastien Le Digabel, 2018. "Computational study of valid inequalities for the maximum k-cut problem," Annals of Operations Research, Springer, vol. 265(1), pages 5-27, June.
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    4. Jamie Fairbrother & Adam N. Letchford & Keith Briggs, 2018. "A two-level graph partitioning problem arising in mobile wireless communications," Computational Optimization and Applications, Springer, vol. 69(3), pages 653-676, April.
    5. Bissan Ghaddar & Miguel Anjos & Frauke Liers, 2011. "A branch-and-cut algorithm based on semidefinite programming for the minimum k-partition problem," Annals of Operations Research, Springer, vol. 188(1), pages 155-174, August.
    6. de Klerk, E. & Pasechnik, D.V. & Warners, J.P., 2004. "On approximate graph colouring and MAX-k-CUT algorithms based on the theta-function," Other publications TiSEM 7a6fbcee-93d0-4f7d-86be-b, Tilburg University, School of Economics and Management.
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