The quadratic assignment problem: the linearization of Xia and Yuan is weaker than the linearization of Adams and Johnson and a family of cuts to narrow the gap

TL;DR

This paper compares two linearizations of the quadratic assignment problem, demonstrating that Adams and Johnson's linearization provides a tighter relaxation than Xia and Yuan's, and introduces ab-cuts to improve Xia and Yuan's formulation.

Contribution

It proves the relative strength of two linearizations and develops a new cutting plane method to enhance Xia and Yuan's linearization.

Findings

Adams and Johnson's linearization has a tighter relaxation than Xia and Yuan's.

Introduction of ab-cuts improves the linearization of Xia and Yuan.

The developed Branch and Cut approach enhances solution quality for the quadratic assignment problem.

Abstract

The quadratic assignment problem is a well-known optimization problem with numerous applications. A common strategy to solve it is to use one of its linearizations and then apply the toolbox of mixed integer linear programming methods. One measure of quality of a mixed integer formulation is the quality of its linear relaxation. In this paper, we compare two linearizations of the quadratic assignment problem and prove that the linear relaxation of the linearization of Adams and Johnson is contained in the linear relaxation of the linearization of Xia and Yuan. We furthermore develop a Branch and Cut approach using the insights obtained in the proof that enhances the linearization of Xia and Yuan via a new family of cuts called ab-cuts.