Estimating Bounds for Quadratic Assignment Problems Associated with Hamming and Manhattan Distance Matrices Based on Semidefinite Programming

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 = ee T 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.

Semi-definite programming relaxation of quadratic assignment problems based on nonredundant matrix splitting

“…To find such a non-redundant matrix splitting, we propose to solve some auxiliary SDP problems following the minimal trace principle 1 . In particular, we show that a straightforward application of the minimal trace principle leads to the so-called orthogonal matrix splitting introduced in [25,28]. We also illustrate that for a given matrix, there may exist multiple non-redundant matrix splitting schemes.…”

“…If the problem (23) is feasible, then the optimal solution α of problem (23) is used to split the matrix B into the following form used in [25]:…”

“…We next present an interesting result regarding problem (25) and the detailed proof of the theorem is given in Appendix 1.…”

“…It is shown that some relaxation models in [25,28] can provide competitive bounds comparing with other relaxation models in the literature. However, since there is a large variety in matrix splitting schemes, it is unclear which splitting scheme can lead to the strongest relaxation.…”

“…Recently, a new class of semi-definite relaxation (SDR) models for QAPs based on matrix splitting has been proposed [25,28]. In this paper, we consider the issue of how to choose an appropriate matrix splitting scheme so that the resulting relaxation model is easy to solve and able to provide a strong bound.…”