A Level-3 Reformulation-Linearization Technique-Based Bound for the Quadratic Assignment Problem

Authors:
Peter M. Hahn;Yi-Rong Zhu;Monique Guignard;William L. Hightower;Matthew J. Saltzman
Affiliations:
Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104;Elder Research, Inc., Charlottesville, Virginia 22903;Operations and Information Management, The Wharton School of the University of Pennsylvania, Philadelphia, Pennsylvania 19104;Mathematics and Computer Science, High Point University, High Point, North Carolina 27262;Mathematical Sciences, Clemson University, Clemson, South Carolina 29634
Venue:
INFORMS Journal on Computing
Year:
2012

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Abstract

We apply the level-3 reformulation-linearization technique (RLT3) to the quadratic assignment problem (QAP). We then present our experience in calculating lower bounds using an essentially new algorithm based on this RLT3 formulation. Our method is not guaranteed to calculate the RLT3 lower bound exactly, but it approximates this lower bound very closely and reaches it in some instances. For Nugent problem instances up to size 24, our RLT3-based lower bound calculation solves these problem instances exactly or serves to verify the optimal value. Calculating lower bounds for problem sizes larger than size 27 still presents a challenge because of the large amount of memory needed to implement the RLT3 formulation. Our presentation emphasizes the steps taken to significantly conserve memory by using the numerous problem symmetries in the RLT3 formulation of the QAP. We implemented this RLT3-based bound calculation in a branch-and-bound algorithm. Experimental results project significant runtime improvement over all other published QAP branch-and-bound solvers.