Graph Modeling for Quadratic Assignment Problems Associated with the Hypercube

Abstract

In the paper we consider the quadratic assignment problem arising from channel coding in communications where one coefficient matrix is the adjacency matrix of a hypercube in a finite dimensional space. By using the geometric structure of the hypercube, we first show that there exist at least n different optimal solutions to the underlying QAPs. Moreover, the inherent symmetries in the associated hypercube allow us to obtain partial information regarding the optimal solutions and thus shrink the search space and improve all the existing QAP solvers for the underlying QAPs.Secondly, we use graph modeling technique to derive a new integer linear program (ILP) models for the underlying QAPs. The new ILP model has n{n -1) binary variables and 0{n log(n)) linear constraints. This yields the smallest known number of binary variables for the ILP reformulation of QAPs. Various relaxations of the new ILP model are obtained based on the graphical characterization of the hypercube, and the lower bounds provided by the LP relaxations of the new model are analyzed and compared with what provided by several classical LP relaxations of QAPs in the literature.

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

“…We mention that in our recent paper [23], we have shown (see Theorem 2.6 of [23]) that for QAPs associated with the hypercube (or Hamming distance matrices), the optimal solution can be attained at a permutation matrix X * with x * 11 = 1. Therefore, we can further add such a constraint to strengthen the bound.…”

“…We next give a technical result that will be used in our later analysis, which is a refinement of Theorem 2.5 in [23].…”

A new relaxation framework for quadratic assignment problems based on matrix splitting

“…We remark that the matrix splitting scheme based on the Laplace operator can also be used to enhance some ILP reformulations of QAPs. For example, when B is the adjacency matrix of a hypercube in a finite space, in [23] we had derived a new ILP reformulation of the original QAP based on the graphical features of the associated hypercube. Since the hypercube is completely connected, we can add a simple SDP constraint to the existing ILP model.…”