IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v40y2020i4d10.1007_s10878-020-00637-6.html
   My bibliography  Save this article

Connected max cut is polynomial for graphs without the excluded minor $$K_5\backslash e$$ K 5 \ e

Author

Listed:
  • Brahim Chaourar

    (Imam Mohammad Ibn Saud Islamic University (IMSIU))

Abstract

Given a graph $$G=(V, E)$$ G = ( V , E ) , a connected cut $$\delta (U)$$ δ ( U ) is the set of edges of E linking all vertices of U to all vertices of $$V\backslash U$$ V \ U such that the induced subgraphs G[U] and $$G[V\backslash U]$$ G [ V \ U ] are connected. Given a positive weight function w defined on E, the connected maximum cut problem (CMAX CUT) is to find a connected cut $$\varOmega $$ Ω such that $$w(\varOmega )$$ w ( Ω ) is maximum among all connected cuts. CMAX CUT is NP-hard even for planar graphs, and thus for graph without the excluded minor $$K_5$$ K 5 . In this paper, we prove that CMAX CUT is polynomial for the class of graphs without the excluded minor $$K_5\backslash e$$ K 5 \ e , denoted by $${\mathcal {G}}(K_5\backslash e)$$ G ( K 5 \ e ) . We deduce two quadratic time algorithms: one for the minimum cut problem in $${\mathcal {G}}(K_5\backslash e)$$ G ( K 5 \ e ) without computing the maximum flow, and another one for Hamilton cycle problem in the class of graphs without the two excluded minors the prism $$P_6$$ P 6 and $$K_{3, 3}$$ K 3 , 3 . This latter problem is NP-complete for maximal planar graphs.

Suggested Citation

  • Brahim Chaourar, 2020. "Connected max cut is polynomial for graphs without the excluded minor $$K_5\backslash e$$ K 5 \ e," Journal of Combinatorial Optimization, Springer, vol. 40(4), pages 869-875, November.
    • Handle: RePEc:spr:jcomop:v:40:y:2020:i:4:d:10.1007_s10878-020-00637-6
    DOI: 10.1007/s10878-020-00637-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-020-00637-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10878-020-00637-6?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.
    Full references (including those not matched with items on IDEAS)

    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. 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. 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.
    3. Goldengorin, Boris, 2009. "Maximization of submodular functions: Theory and enumeration algorithms," European Journal of Operational Research, Elsevier, vol. 198(1), pages 102-112, October.
    4. 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.
    5. 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.
    6. 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.
    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. 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.
    10. 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.
    11. 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).
    12. 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.
    13. 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).
    14. 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.
    15. 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.
    16. 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.
    17. 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.
    18. 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.
    19. 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.
    20. 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.
    21. 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).

    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:jcomop:v:40:y:2020:i:4:d:10.1007_s10878-020-00637-6. 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.