Exactness Conditions for Semidefinite Relaxations of the Quadratic Assignment Problem

TL;DR

This paper establishes simple deterministic conditions under which semidefinite relaxations of the NP-hard Quadratic Assignment Problem are exact, providing a foundation for future research and practical solution methods.

Contribution

It introduces a set of linear inequalities that guarantee the exactness of semidefinite relaxations for the QAP, and demonstrates their applicability under small perturbations.

Findings

Semidefinite relaxations are exact under specific linear inequality conditions.

Exactness holds under small perturbations of input matrices.

Constructs a sequence of dual feasible solutions approaching optimality.

Abstract

The Quadratic Assignment Problem (QAP) is an important discrete optimization instance that encompasses many well-known combinatorial optimization problems, and has applications in a wide range of areas such as logistics and computer vision. The QAP, unfortunately, is NP-hard to solve. To address this difficulty, a number of semidefinite relaxations of the QAP have been developed. These techniques are known to be powerful in that they compute globally optimal solutions in many instances, and are often deployed as sub-routines within enumerative procedures for solving QAPs. In this paper, we investigate the strength of these semidefinite relaxations. Our main result is a deterministic set of conditions on the input matrices -- specified via linear inequalities -- under which these semidefinite relaxations are exact. Our result is simple to state, in the hope that it serves as a foundation for subsequent studies on semidefinite relaxations of the QAP as well as other related combinatorial optimization problems. As a corollary, we show that the semidefinite relaxation is exact under a perturbation model whereby the input matrices differ by a suitably small perturbation term. One technical difficulty we encounter is that the set of dual feasible solutions is not closed. To circumvent these difficulties, our analysis constructs a sequence of dual feasible solutions whose objective value approaches the primal objective, but never attains it.