IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v188y2011i1p155-17410.1007-s10479-008-0481-4.html
   My bibliography  Save this article

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

Author

Listed:
  • 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

Suggested Citation

  • 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
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10479-008-0481-4
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10479-008-0481-4?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. John E. Mitchell, 2003. "Realignment in the National Football League: Did they do it right?," Naval Research Logistics (NRL), John Wiley & Sons, vol. 50(7), pages 683-701, October.
    2. 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.
    3. Francisco Barahona & Martin Grötschel & Michael Jünger & Gerhard Reinelt, 1988. "An Application of Combinatorial Optimization to Statistical Physics and Circuit Layout Design," Operations Research, INFORMS, vol. 36(3), pages 493-513, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. 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.
    8. 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.
    9. 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.
    10. 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.
    11. 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.

    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. 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.
    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.
    3. 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.
    4. Wenxing Zhu & Geng Lin & M. M. Ali, 2013. "Max- k -Cut by the Discrete Dynamic Convexized Method," INFORMS Journal on Computing, INFORMS, vol. 25(1), pages 27-40, February.
    5. 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.
    6. Dell'Amico, Mauro & Trubian, Marco, 1998. "Solution of large weighted equicut problems," European Journal of Operational Research, Elsevier, vol. 106(2-3), pages 500-521, April.
    7. Shenshen Gu & Yue Yang, 2020. "A Deep Learning Algorithm for the Max-Cut Problem Based on Pointer Network Structure with Supervised Learning and Reinforcement Learning Strategies," Mathematics, MDPI, vol. 8(2), pages 1-20, February.
    8. repec:dgr:rugsom:99a17 is not listed on IDEAS
    9. Gili Rosenberg & Mohammad Vazifeh & Brad Woods & Eldad Haber, 2016. "Building an iterative heuristic solver for a quantum annealer," Computational Optimization and Applications, Springer, vol. 65(3), pages 845-869, December.
    10. Pierre Fouilhoux & A. Mahjoub, 2012. "Solving VLSI design and DNA sequencing problems using bipartization of graphs," Computational Optimization and Applications, Springer, vol. 51(2), pages 749-781, March.
    11. Goldengorin, Boris & Ghosh, Diptesh, 2004. "A Multilevel Search Algorithm for the Maximization of Submodular Functions," Research Report 04A20, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    12. Hongwei Liu & Sanyang Liu & Fengmin Xu, 2003. "A Tight Semidefinite Relaxation of the MAX CUT Problem," Journal of Combinatorial Optimization, Springer, vol. 7(3), pages 237-245, September.
    13. Daya Ram Gaur & Ramesh Krishnamurti & Rajeev Kohli, 2009. "Conflict Resolution in the Scheduling of Television Commercials," Operations Research, INFORMS, vol. 57(5), pages 1098-1105, October.
    14. Goldengorin, Boris & Tijssen, Gert A. & Tso, Michael, 1999. "The maximization of submodular functions : old and new proofs for the correctness of the dichotomy algorithm," Research Report 99A17, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    15. repec:dgr:rugsom:98a08 is not listed on IDEAS
    16. Ling, Ai-Fan & Xu, Cheng-Xian & Xu, Feng-Min, 2009. "A discrete filled function algorithm embedded with continuous approximation for solving max-cut problems," European Journal of Operational Research, Elsevier, vol. 197(2), pages 519-531, September.
    17. F. Rendl, 2016. "Semidefinite relaxations for partitioning, assignment and ordering problems," Annals of Operations Research, Springer, vol. 240(1), pages 119-140, May.
    18. Mansouri, Bahareh & Hassini, Elkafi, 2019. "Optimal pricing in iterative flexible combinatorial procurement auctions," European Journal of Operational Research, Elsevier, vol. 277(3), pages 1083-1097.
    19. Chuangchuang Sun, 2023. "A Customized ADMM Approach for Large-Scale Nonconvex Semidefinite Programming," Mathematics, MDPI, vol. 11(21), pages 1-27, October.
    20. Kuryatnikova, Olga & Sotirov, Renata & Vera, J.C., 2022. "The maximum $k$-colorable subgraph problem and related problems," Other publications TiSEM 40e477c0-a78e-4ee1-92de-8, Tilburg University, School of Economics and Management.
    21. F. Liers & G. Pardella, 2012. "Partitioning planar graphs: a fast combinatorial approach for max-cut," Computational Optimization and Applications, Springer, vol. 51(1), pages 323-344, January.
    22. Chao Yun & Zhongyu Liang & Aleš Hrabec & Zhentao Liu & Mantao Huang & Leran Wang & Yifei Xiao & Yikun Fang & Wei Li & Wenyun Yang & Yanglong Hou & Jinbo Yang & Laura J. Heyderman & Pietro Gambardella , 2023. "Electrically programmable magnetic coupling in an Ising network exploiting solid-state ionic gating," Nature Communications, Nature, vol. 14(1), pages 1-9, December.

    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:spr:annopr:v:188:y:2011:i:1:p:155-174:10.1007/s10479-008-0481-4. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.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.