A Low-Dimensional Semidefinite Relaxation for the Quadratic Assignment Problem

Authors:
Yichuan Ding;Henry Wolkowicz
Affiliations:
Department of Management Science and Engineering, School of Engineering, Stanford University, Stanford, California 94305;Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Venue:
Mathematics of Operations Research
Year:
2009

Citing 14
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Abstract

The quadratic assignment problem (QAP) is arguably one of the hardest NP-hard discrete optimization problems. Problems of dimension greater than 25 are still considered to be large scale. Current successful solution techniques use branch-and-bound methods, which rely on obtaining strong and inexpensive bounds. In this paper, we introduce a new semidefinite programming (SDP) relaxation for generating bounds for the QAP in the trace formulation. We apply majorization to obtain a relaxation of the orthogonal similarity set of the quadratic part of the objective function. This exploits the matrix structure of QAP and results in a relaxation with much smaller dimension than other current SDP relaxations. We compare the resulting bounds with several other computationally inexpensive bounds such as the convex quadratic programming relaxation (QPB). We find that our method provides stronger bounds on average and is adaptable for branch-and-bound methods.