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

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 ex...

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Bibliographic Details
Published in:INFORMS journal on computing 2012-03, Vol.24 (2), p.202-209
Main Authors: Hahn, Peter M, Zhu, Yi-Rong, Guignard, Monique, Hightower, William L, Saltzman, Matthew J
Format: Article
Language:English
Subjects:
Algorithms
Assignment problem
Branch & bound algorithms
Combinatorial analysis
Costs
dual ascent
exact solution
Integer programming
Linear equations
Linear programming
Mathematical optimization
QAP
quadratic assignment
Quadratic programming
reformulation linearization
RLT
Studies
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Summary: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.
ISSN:1091-9856
1526-5528
1091-9856
DOI:10.1287/ijoc.1110.0450