On the Complexity of Some Facet-Defining Inequalities of the QAP-polytope

TL;DR

This paper investigates the complexity of facet-defining inequalities of the QAP polytope, introduces a new family of such inequalities, and proves the computational hardness of membership testing for some classes, highlighting the problem's intrinsic difficulty.

Contribution

It introduces a new family of facet-defining inequalities for the QAP polytope and establishes the coNP-completeness of membership testing for certain inequality classes.

Findings

New family of facet-defining inequalities identified

Membership testing for some inequalities is coNP-complete

Lower bound of exponential extension complexity for relaxations

Abstract

The Quadratic Assignment Problem (QAP) is a well-known NP-hard problem that is equivalent to optimizing a linear objective function over the QAP polytope. The QAP polytope with parameter $n$ - \qappolytope{n} - is defined as the convex hull of rank-$1$ matrices $xx^T$ with $x$ as the vectorized $n\times n$ permutation matrices. In this paper we consider all the known exponential-sized families of facet-defining inequalities of the QAP-polytope. We describe a new family of valid inequalities that we show to be facet-defining. We also show that membership testing (and hence optimizing) over some of the known classes of inequalities is coNP-complete. We complement our hardness results by showing a lower bound of $2^{\Omega(n)}$ on the extension complexity of all relaxations of \qappolytope{n} for which any of the known classes of inequalities are valid.