IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v69y2018i3d10.1007_s10589-017-9968-8.html
   My bibliography  Save this article

Semidefinite programming approach for the quadratic assignment problem with a sparse graph

Author

Listed:
  • José F. S. Bravo Ferreira

    (Princeton University)

  • Yuehaw Khoo

    (Stanford University)

  • Amit Singer

    (Princeton University)

Abstract

The matching problem between two adjacency matrices can be formulated as the NP-hard quadratic assignment problem (QAP). Previous work on semidefinite programming (SDP) relaxations to the QAP have produced solutions that are often tight in practice, but such SDPs typically scale badly, involving matrix variables of dimension $$n^2$$ n 2 where n is the number of nodes. To achieve a speed up, we propose a further relaxation of the SDP involving a number of positive semidefinite matrices of dimension $$\mathcal {O}(n)$$ O ( n ) no greater than the number of edges in one of the graphs. The relaxation can be further strengthened by considering cliques in the graph, instead of edges. The dual problem of this novel relaxation has a natural three-block structure that can be solved via a convergent Alternating Direction Method of Multipliers in a distributed manner, where the most expensive step per iteration is computing the eigendecomposition of matrices of dimension $$\mathcal {O}(n)$$ O ( n ) . The new SDP relaxation produces strong bounds on quadratic assignment problems where one of the graphs is sparse with reduced computational complexity and running times, and can be used in the context of nuclear magnetic resonance spectroscopy to tackle the assignment problem.

Suggested Citation

  • José F. S. Bravo Ferreira & Yuehaw Khoo & Amit Singer, 2018. "Semidefinite programming approach for the quadratic assignment problem with a sparse graph," Computational Optimization and Applications, Springer, vol. 69(3), pages 677-712, April.
    • Handle: RePEc:spr:coopap:v:69:y:2018:i:3:d:10.1007_s10589-017-9968-8
    DOI: 10.1007/s10589-017-9968-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-017-9968-8
    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/s10589-017-9968-8?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. 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.
    2. Tjalling C. Koopmans & Martin J. Beckmann, 1955. "Assignment Problems and the Location of Economic Activities," Cowles Foundation Discussion Papers 4, Cowles Foundation for Research in Economics, Yale University.
    3. de Klerk, E. & Sotirov, R., 2007. "Exploiting Group Symmetry in Semidefinite Programming Relaxations of the Quadratic Assignment Problem," Discussion Paper 2007-44, Tilburg University, Center for Economic Research.
    4. Qing Zhao & Stefan E. Karisch & Franz Rendl & Henry Wolkowicz, 1998. "Semidefinite Programming Relaxations for the Quadratic Assignment Problem," Journal of Combinatorial Optimization, Springer, vol. 2(1), pages 71-109, March.
    5. Nicos Christofides & Enrique Benavent, 1989. "An Exact Algorithm for the Quadratic Assignment Problem on a Tree," Operations Research, INFORMS, vol. 37(5), pages 760-768, October.
    6. Jiming Peng & Tao Zhu & Hezhi Luo & Kim-Chuan Toh, 2015. "Semi-definite programming relaxation of quadratic assignment problems based on nonredundant matrix splitting," Computational Optimization and Applications, Springer, vol. 60(1), pages 171-198, January.
    7. E. de Klerk & R. Sotirov & U. Truetsch, 2015. "A New Semidefinite Programming Relaxation for the Quadratic Assignment Problem and Its Computational Perspectives," INFORMS Journal on Computing, INFORMS, vol. 27(2), pages 378-391, May.
    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. José F. S. Bravo-Ferreira & David Cowburn & Yuehaw Khoo & Amit Singer, 2022. "NMR assignment through linear programming," Journal of Global Optimization, Springer, vol. 83(1), pages 3-28, May.
    2. Naomi Graham & Hao Hu & Jiyoung Im & Xinxin Li & Henry Wolkowicz, 2022. "A Restricted Dual Peaceman-Rachford Splitting Method for a Strengthened DNN Relaxation for QAP," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 2125-2143, July.

    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. Jiming Peng & Tao Zhu & Hezhi Luo & Kim-Chuan Toh, 2015. "Semi-definite programming relaxation of quadratic assignment problems based on nonredundant matrix splitting," Computational Optimization and Applications, Springer, vol. 60(1), pages 171-198, January.
    2. E. de Klerk & R. Sotirov & U. Truetsch, 2015. "A New Semidefinite Programming Relaxation for the Quadratic Assignment Problem and Its Computational Perspectives," INFORMS Journal on Computing, INFORMS, vol. 27(2), pages 378-391, May.
    3. Feizollahi, Mohammad Javad & Feyzollahi, Hadi, 2015. "Robust quadratic assignment problem with budgeted uncertain flows," Operations Research Perspectives, Elsevier, vol. 2(C), pages 114-123.
    4. Yichuan Ding & Dongdong Ge & Henry Wolkowicz, 2011. "On Equivalence of Semidefinite Relaxations for Quadratic Matrix Programming," Mathematics of Operations Research, INFORMS, vol. 36(1), pages 88-104, February.
    5. Yong Xia & Wajeb Gharibi, 2015. "On improving convex quadratic programming relaxation for the quadratic assignment problem," Journal of Combinatorial Optimization, Springer, vol. 30(3), pages 647-667, October.
    6. Nyberg, Axel & Westerlund, Tapio, 2012. "A new exact discrete linear reformulation of the quadratic assignment problem," European Journal of Operational Research, Elsevier, vol. 220(2), pages 314-319.
    7. Nihal Berktaş & Hande Yaman, 2021. "A Branch-and-Bound Algorithm for Team Formation on Social Networks," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 1162-1176, July.
    8. Boukouvala, Fani & Misener, Ruth & Floudas, Christodoulos A., 2016. "Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO," European Journal of Operational Research, Elsevier, vol. 252(3), pages 701-727.
    9. 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.
    10. de Klerk, E. & Pasechnik, D.V. & Sotirov, R., 2007. "On Semidefinite Programming Relaxations of the Travelling Salesman Problem (Replaced by DP 2008-96)," Discussion Paper 2007-101, Tilburg University, Center for Economic Research.
    11. E. R. van Dam & R. Sotirov, 2015. "On Bounding the Bandwidth of Graphs with Symmetry," INFORMS Journal on Computing, INFORMS, vol. 27(1), pages 75-88, February.
    12. de Klerk, Etienne & -Nagy, Marianna E. & Sotirov, Renata & Truetsch, Uwe, 2014. "Symmetry in RLT-type relaxations for the quadratic assignment and standard quadratic optimization problems," European Journal of Operational Research, Elsevier, vol. 233(3), pages 488-499.
    13. Ketan Date & Rakesh Nagi, 2019. "Level 2 Reformulation Linearization Technique–Based Parallel Algorithms for Solving Large Quadratic Assignment Problems on Graphics Processing Unit Clusters," INFORMS Journal on Computing, INFORMS, vol. 31(4), pages 771-789, October.
    14. Christoph Buchheim & Emiliano Traversi, 2018. "Quadratic Combinatorial Optimization Using Separable Underestimators," INFORMS Journal on Computing, INFORMS, vol. 30(3), pages 424-437, August.
    15. de Klerk, E. & Pasechnik, D.V. & Sotirov, R., 2008. "On Semidefinite Programming Relaxations of the Traveling Salesman Problem (revision of DP 2007-101)," Discussion Paper 2008-96, Tilburg University, Center for Economic Research.
    16. Fanz Rendl & Renata Sotirov, 2018. "The min-cut and vertex separator problem," Computational Optimization and Applications, Springer, vol. 69(1), pages 159-187, January.
    17. Hu, Hao & Sotirov, Renata & Wolkowicz, Henry, 2023. "Facial reduction for symmetry reduced semidefinite and doubly nonnegative programs," Other publications TiSEM 8dd3dbae-58fd-4238-b786-e, Tilburg University, School of Economics and Management.
    18. Brosch, Daniel & de Klerk, Etienne, 2021. "Jordan symmetry reduction for conic optimization over the doubly nonnegative cone: Theory and software," Other publications TiSEM 283da78a-b42f-47b4-b2b7-2, Tilburg University, School of Economics and Management.
    19. Alcaide-López-de-Pablo, David & Sicilia, Joaquín & González-Sierra, Miguel Á., 2017. "Locating names on vertices of a transaction network," European Journal of Operational Research, Elsevier, vol. 262(2), pages 464-478.
    20. Ting Pong & Hao Sun & Ningchuan Wang & Henry Wolkowicz, 2016. "Eigenvalue, quadratic programming, and semidefinite programming relaxations for a cut minimization problem," Computational Optimization and Applications, Springer, vol. 63(2), pages 333-364, March.

    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:coopap:v:69:y:2018:i:3:d:10.1007_s10589-017-9968-8. 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.