TY - JOUR

T1 - Semidefinite programming approach for the quadratic assignment problem with a sparse graph

AU - Bravo Ferreira, José F.S.

AU - Khoo, Yuehaw

AU - Singer, Amit

N1 - Funding Information:
Acknowledgements The authors would like to thank David Cowburn for useful discussions on NMR spectroscopy, and Amir Ali Ahmadi, for suggestions on tightening the SDP relaxations presented here. The authors would also like to thank the anonymous reviewers for their useful feedback and suggestions. The authors were partially supported by Award Number R01GM090200 from the NIGMS, FA9550-12-1-0317 from AFOSR, the Simons Investigator Award and the Simons Collaboration on Algorithms and Geometry from Simons Foundation, and the Moore Foundation Data-Driven Discovery Investigator Award.
Publisher Copyright:
© 2017, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2018/4/1

Y1 - 2018/4/1

N2 - 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 n2 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 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 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.

AB - 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 n2 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 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 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.

KW - Alternating direction method of multipliers

KW - Convex relaxation

KW - Graph matching

KW - Quadratic assignment problem

KW - Semidefinite programming

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U2 - 10.1007/s10589-017-9968-8

DO - 10.1007/s10589-017-9968-8

M3 - Article

AN - SCOPUS:85034241306

VL - 69

SP - 677

EP - 712

JO - Computational Optimization and Applications

JF - Computational Optimization and Applications

SN - 0926-6003

IS - 3

ER -