Estimating Bounds for Quadratic Assignment Problems Associated with Hamming and Manhattan Distance Matrices based on Semidefinite Programming Hans Mittelmann∗ Jiming Peng
†
September 17, 2010 Abstract Quadratic assignment problems (QAPs) with a Hamming distance matrix for a hypercube or a Manhattan distance matrix for a rectangular grid arise frequently from communications and facility locations and are known to be among the hardest discrete optimization problems. In this paper we consider the issue of how to obtain lower bounds for those two classes of QAPs based on semidefinite programming (SDP). By exploiting the data structure of the distance matrix B, we first show that for any permutation matrix X, the matrix Y = αE − XBX T is positive semi-definite, where α is a properly chosen parameter depending only on the associated graph in the underlying QAP and E = eeT is a rank one matrix whose elements have value 1. This results in a natural way to approximate the original QAPs via SDP relaxation based on the matrix splitting technique. Our new SDP relaxations have a smaller size compared with other SDP relaxations in the literature and can be solved efficiently by most open source SDP solvers. Experimental results show that for the underlying QAPs of size up to n=200, strong bounds can be obtained effectively.
Key words. Quadratic Assignment Problem (QAP), Semidefinite Programming (SDP), Singular Value Decomposition (SVD), Relaxation, Lower Bound.
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Introduction
The standard quadratic assignment problem takes the following form min Tr AXBX T X∈Π
(1)
∗ Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 852871804, USA. Email:
[email protected]. † Corresponding author. Department of Industrial and Enterprise System Engineering , University of Illinois at Urbana-Champaign. Urbana, IL, 61801.
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where A, B ∈
