IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v54y2012i2p341-351.html
   My bibliography  Save this article

Canonical dual approach to solving the maximum cut problem

Author

Listed:
  • Zhenbo Wang
  • Shu-Cherng Fang
  • David Gao
  • Wenxun Xing

Abstract

This paper presents a canonical dual approach for finding either an optimal or approximate solution to the maximum cut problem (MAX CUT). We show that, by introducing a linear perturbation term to the objective function, the maximum cut problem is perturbed to have a dual problem which is a concave maximization problem over a convex feasible domain under certain conditions. Consequently, some global optimality conditions are derived for finding an optimal or approximate solution. A gradient decent algorithm is proposed for this purpose and computational examples are provided to illustrate the proposed approach. Copyright Springer Science+Business Media, LLC. 2012

Suggested Citation

  • Zhenbo Wang & Shu-Cherng Fang & David Gao & Wenxun Xing, 2012. "Canonical dual approach to solving the maximum cut problem," Journal of Global Optimization, Springer, vol. 54(2), pages 341-351, October.
    • Handle: RePEc:spr:jglopt:v:54:y:2012:i:2:p:341-351
    DOI: 10.1007/s10898-012-9881-8
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10898-012-9881-8
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10898-012-9881-8?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. 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.
    2. David Gao & Ning Ruan, 2010. "Solutions to quadratic minimization problems with box and integer constraints," Journal of Global Optimization, Springer, vol. 47(3), pages 463-484, July.
    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. Yi Chen & David Gao, 2016. "Global solutions to nonconvex optimization of 4th-order polynomial and log-sum-exp functions," Journal of Global Optimization, Springer, vol. 64(3), pages 417-431, March.
    2. Vittorio Latorre & Simone Sagratella, 2016. "A canonical duality approach for the solution of affine quasi-variational inequalities," Journal of Global Optimization, Springer, vol. 64(3), pages 433-449, March.

    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. Gary Kochenberger & Jin-Kao Hao & Fred Glover & Mark Lewis & Zhipeng Lü & Haibo Wang & Yang Wang, 2014. "The unconstrained binary quadratic programming problem: a survey," Journal of Combinatorial Optimization, Springer, vol. 28(1), pages 58-81, July.
    2. 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.
    3. 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.
    4. Goldengorin, Boris, 2009. "Maximization of submodular functions: Theory and enumeration algorithms," European Journal of Operational Research, Elsevier, vol. 198(1), pages 102-112, October.
    5. Xunzhao Yin & Yu Qian & Alptekin Vardar & Marcel Günther & Franz Müller & Nellie Laleni & Zijian Zhao & Zhouhang Jiang & Zhiguo Shi & Yiyu Shi & Xiao Gong & Cheng Zhuo & Thomas Kämpfe & Kai Ni, 2024. "Ferroelectric compute-in-memory annealer for combinatorial optimization problems," Nature Communications, Nature, vol. 15(1), pages 1-11, December.
    6. 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.
    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. Jin, Zhong & Y. Gao, David, 2017. "On modeling and global solutions for d.c. optimization problems by canonical duality theory," Applied Mathematics and Computation, Elsevier, vol. 296(C), pages 168-181.
    9. repec:dgr:rugsom:99a17 is not listed on IDEAS
    10. Fred Glover & Jin-Kao Hao, 2016. "f-Flip strategies for unconstrained binary quadratic programming," Annals of Operations Research, Springer, vol. 238(1), pages 651-657, March.
    11. 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.
    12. Goldengorin, Boris & Sierksma, Gerard & Tijssen, Gert A., 1998. "The data-correcting algorithm for supermodular functions, with applications to quadratic cost partition and simple plant location problems," Research Report 98A08, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    13. 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.
    14. 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).
    15. Iain Dunning & Swati Gupta & John Silberholz, 2018. "What Works Best When? A Systematic Evaluation of Heuristics for Max-Cut and QUBO," INFORMS Journal on Computing, INFORMS, vol. 30(3), pages 608-624, August.
    16. 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.
    17. Aardal, K.I. & van Hoesel, S., 1995. "Polyhedral Techniques in Combinatorial Optimization," Other publications TiSEM ed028a07-eb6a-4c8d-8f21-d, Tilburg University, School of Economics and Management.
    18. 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.
    19. Arman Boyacı & Tınaz Ekim & Mordechai Shalom, 2018. "The maximum cardinality cut problem in co-bipartite chain graphs," Journal of Combinatorial Optimization, Springer, vol. 35(1), pages 250-265, January.
    20. Boris Goldengorin & Gerard Sierksma & Gert A. Tijssen & Michael Tso, 1999. "The Data-Correcting Algorithm for the Minimization of Supermodular Functions," Management Science, INFORMS, vol. 45(11), pages 1539-1551, November.
    21. Geng Lin & Wenxing Zhu, 2012. "A discrete dynamic convexized method for the max-cut problem," Annals of Operations Research, Springer, vol. 196(1), pages 371-390, July.

    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:jglopt:v:54:y:2012:i:2:p:341-351. 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.