IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v39y2020i4d10.1007_s10878-020-00547-7.html
   My bibliography  Save this article

The quadratic cycle cover problem: special cases and efficient bounds

Author

Listed:
  • Frank Meijer

    (Tilburg University)

  • Renata Sotirov

    (Tilburg University)

Abstract

The quadratic cycle cover problem is the problem of finding a set of node-disjoint cycles visiting all the nodes such that the total sum of interaction costs between consecutive arcs is minimized. In this paper we study the linearization problem for the quadratic cycle cover problem and related lower bounds. In particular, we derive various sufficient conditions for the quadratic cost matrix to be linearizable, and use these conditions to compute bounds. We also show how to use a sufficient condition for linearizability within an iterative bounding procedure. In each step, our algorithm computes the best equivalent representation of the quadratic cost matrix and its optimal linearizable matrix with respect to the given sufficient condition for linearizability. Further, we show that the classical Gilmore–Lawler type bound belongs to the family of linearization based bounds, and therefore apply the above mentioned iterative reformulation technique. We also prove that the linearization vectors resulting from this iterative approach satisfy the constant value property. The best among here introduced bounds outperform existing lower bounds when taking both quality and efficiency into account.

Suggested Citation

  • Frank Meijer & Renata Sotirov, 2020. "The quadratic cycle cover problem: special cases and efficient bounds," Journal of Combinatorial Optimization, Springer, vol. 39(4), pages 1096-1128, May.
    • Handle: RePEc:spr:jcomop:v:39:y:2020:i:4:d:10.1007_s10878-020-00547-7
    DOI: 10.1007/s10878-020-00547-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-020-00547-7
    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/s10878-020-00547-7?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. Paolo Carraresi & Federico Malucelli, 1992. "A New Lower Bound for the Quadratic Assignment Problem," Operations Research, INFORMS, vol. 40(1-supplem), pages 22-27, February.
    2. Fred Glover, 1975. "Improved Linear Integer Programming Formulations of Nonlinear Integer Problems," Management Science, INFORMS, vol. 22(4), pages 455-460, December.
    3. Hao Hu & Renata Sotirov, 2018. "Special cases of the quadratic shortest path problem," Journal of Combinatorial Optimization, Springer, vol. 35(3), pages 754-777, April.
    4. Ante Ćustić & Abraham P. Punnen, 2018. "A characterization of linearizable instances of the quadratic minimum spanning tree problem," Journal of Combinatorial Optimization, Springer, vol. 35(2), pages 436-453, February.
    5. Rostami, Borzou & Chassein, André & Hopf, Michael & Frey, Davide & Buchheim, Christoph & Malucelli, Federico & Goerigk, Marc, 2018. "The quadratic shortest path problem: complexity, approximability, and solution methods," European Journal of Operational Research, Elsevier, vol. 268(2), pages 473-485.
    6. Santosh N. Kabadi & Abraham P. Punnen, 2011. "An O ( n 4 ) Algorithm for the QAP Linearization Problem," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 754-761, November.
    7. Eugene L. Lawler, 1963. "The Quadratic Assignment Problem," Management Science, INFORMS, vol. 9(4), pages 586-599, 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. Frank de Meijer & Renata Sotirov, 2021. "SDP-Based Bounds for the Quadratic Cycle Cover Problem via Cutting-Plane Augmented Lagrangian Methods and Reinforcement Learning," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1262-1276, October.
    2. Hao Hu & Renata Sotirov, 2021. "The linearization problem of a binary quadratic problem and its applications," Annals of Operations Research, Springer, vol. 307(1), pages 229-249, December.

    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. Hao Hu & Renata Sotirov, 2021. "The linearization problem of a binary quadratic problem and its applications," Annals of Operations Research, Springer, vol. 307(1), pages 229-249, December.
    2. Rostami, Borzou & Chassein, André & Hopf, Michael & Frey, Davide & Buchheim, Christoph & Malucelli, Federico & Goerigk, Marc, 2018. "The quadratic shortest path problem: complexity, approximability, and solution methods," European Journal of Operational Research, Elsevier, vol. 268(2), pages 473-485.
    3. de Meijer, Frank, 2023. "Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization," Other publications TiSEM b1f1088c-95fe-4b8a-9e15-c, Tilburg University, School of Economics and Management.
    4. Vittorio Maniezzo, 1999. "Exact and Approximate Nondeterministic Tree-Search Procedures for the Quadratic Assignment Problem," INFORMS Journal on Computing, INFORMS, vol. 11(4), pages 358-369, November.
    5. Mans, Bernard & Mautor, Thierry & Roucairol, Catherine, 1995. "A parallel depth first search branch and bound algorithm for the quadratic assignment problem," European Journal of Operational Research, Elsevier, vol. 81(3), pages 617-628, March.
    6. Yang Wang & Wei Yang & Abraham P. Punnen & Jingbo Tian & Aihua Yin & Zhipeng Lü, 2021. "The Rank-One Quadratic Assignment Problem," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 979-996, July.
    7. Christoph Buchheim & Emiliano Traversi, 2018. "Quadratic Combinatorial Optimization Using Separable Underestimators," INFORMS Journal on Computing, INFORMS, vol. 30(3), pages 424-437, August.
    8. Ball, Michael O. & Kaku, Bharat K. & Vakhutinsky, Andrew, 1998. "Network-based formulations of the quadratic assignment problem," European Journal of Operational Research, Elsevier, vol. 104(1), pages 241-249, January.
    9. Mamadou Koné & Mouhamadou A.M.T. Baldé & Babacar M. Ndiaye, 2019. "A Dichotomic Algorithm for Transportation Network and Land Use Problem," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 11(1), pages 42-56, February.
    10. Solimanpur, M. & Vrat, P. & Shankar, R., 2004. "Ant colony optimization algorithm to the inter-cell layout problem in cellular manufacturing," European Journal of Operational Research, Elsevier, vol. 157(3), pages 592-606, September.
    11. Zhang, Yufeng & Khani, Alireza, 2019. "An algorithm for reliable shortest path problem with travel time correlations," Transportation Research Part B: Methodological, Elsevier, vol. 121(C), pages 92-113.
    12. Loiola, Eliane Maria & de Abreu, Nair Maria Maia & Boaventura-Netto, Paulo Oswaldo & Hahn, Peter & Querido, Tania, 2007. "A survey for the quadratic assignment problem," European Journal of Operational Research, Elsevier, vol. 176(2), pages 657-690, January.
    13. David Bergman, 2019. "An Exact Algorithm for the Quadratic Multiknapsack Problem with an Application to Event Seating," INFORMS Journal on Computing, INFORMS, vol. 31(3), pages 477-492, July.
    14. Brad D. Woods & Abraham P. Punnen, 0. "A class of exponential neighbourhoods for the quadratic travelling salesman problem," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-30.
    15. Hahn, Peter & Grant, Thomas & Hall, Nat, 1998. "A branch-and-bound algorithm for the quadratic assignment problem based on the Hungarian method," European Journal of Operational Research, Elsevier, vol. 108(3), pages 629-640, August.
    16. Brad D. Woods & Abraham P. Punnen, 2020. "A class of exponential neighbourhoods for the quadratic travelling salesman problem," Journal of Combinatorial Optimization, Springer, vol. 40(2), pages 303-332, August.
    17. Jingyang Zhou & Peter E.D. Love & Kok Lay Teo & Hanbin Luo, 2017. "An exact penalty function method for optimising QAP formulation in facility layout problem," International Journal of Production Research, Taylor & Francis Journals, vol. 55(10), pages 2913-2929, May.
    18. Yokoyama, Ryohei & Kitano, Hiroyuki & Wakui, Tetsuya, 2017. "Optimal operation of heat supply systems with piping network," Energy, Elsevier, vol. 137(C), pages 888-897.
    19. Tian, Xueyu & You, Fengqi, 2019. "Carbon-neutral hybrid energy systems with deep water source cooling, biomass heating, and geothermal heat and power," Applied Energy, Elsevier, vol. 250(C), pages 413-432.
    20. Mohammad Javad Feizollahi & Igor Averbakh, 2014. "The Robust (Minmax Regret) Quadratic Assignment Problem with Interval Flows," INFORMS Journal on Computing, INFORMS, vol. 26(2), pages 321-335, May.

    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:39:y:2020:i:4:d:10.1007_s10878-020-00547-7. 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.