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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
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    References listed on IDEAS

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    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. 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.
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    5. 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.
    6. 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.
    7. 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.
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    Cited by:

    1. 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.
    2. 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.

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