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A branch-and-cut algorithm based on semidefinite programming for the minimum k-partition problem

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  • Bissan Ghaddar
  • Miguel Anjos
  • Frauke Liers

Abstract

The minimum k-partition (MkP) problem is the problem of partitioning the set of vertices of a graph into k disjoint subsets so as to minimize the total weight of the edges joining vertices in the same partition. The main contribution of this paper is the design and implementation of a branch-and-cut algorithm based on semidefinite programming (SBC) for the MkP problem. The two key ingredients for this algorithm are: the combination of semidefinite programming with polyhedral results; and a novel iterative clustering heuristic (ICH) that finds feasible solutions for the MkP problem. We compare ICH to the hyperplane rounding techniques of Goemans and Williamson and of Frieze and Jerrum, and the computational results support the conclusion that ICH consistently provides better feasible solutions for the MkP problem. ICH is used in our SBC algorithm to provide feasible solutions at each node of the branch-and-bound tree. The SBC algorithm computes globally optimal solutions for dense graphs with up to 60 vertices, for grid graphs with up to 100 vertices, and for different values of k, providing a fast exact approach for k≥3. Copyright Springer Science+Business Media, LLC 2011

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  • 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.
    • Handle: RePEc:spr:annopr:v:188:y:2011:i:1:p:155-174:10.1007/s10479-008-0481-4
    DOI: 10.1007/s10479-008-0481-4
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    Cited by:

    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. Cheng Lu & Zhibin Deng, 2021. "A branch-and-bound algorithm for solving max-k-cut problem," Journal of Global Optimization, Springer, vol. 81(2), pages 367-389, October.
    3. Angelika Wiegele & Shudian Zhao, 2022. "SDP-based bounds for graph partition via extended ADMM," Computational Optimization and Applications, Springer, vol. 82(1), pages 251-291, May.
    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. F. Rendl, 2016. "Semidefinite relaxations for partitioning, assignment and ordering problems," Annals of Operations Research, Springer, vol. 240(1), pages 119-140, May.
    6. Joe Naoum-Sawaya & Christoph Buchheim, 2016. "Robust Critical Node Selection by Benders Decomposition," INFORMS Journal on Computing, INFORMS, vol. 28(1), pages 162-174, February.
    7. Brais González-Rodríguez & Raúl Alvite-Pazó & Samuel Alvite-Pazó & Bissan Ghaddar & Julio González-Díaz, 2025. "Polynomial Optimization: Tightening RLT-Based Branch-and-Bound Schemes with Conic Constraints," Journal of Optimization Theory and Applications, Springer, vol. 204(1), pages 1-34, January.
    8. 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.
    9. Giovanni Giallombardo & Houyuan Jiang & Giovanna Miglionico, 2016. "New Formulations for the Conflict Resolution Problem in the Scheduling of Television Commercials," Operations Research, INFORMS, vol. 64(4), pages 838-848, August.
    10. Veronica Piccialli & Antonio M. Sudoso & Angelika Wiegele, 2022. "SOS-SDP: An Exact Solver for Minimum Sum-of-Squares Clustering," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 2144-2162, July.
    11. Renata Sotirov, 2014. "An Efficient Semidefinite Programming Relaxation for the Graph Partition Problem," INFORMS Journal on Computing, INFORMS, vol. 26(1), pages 16-30, February.
    12. 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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