New Facets of the QAP-Polytope

TL;DR

This paper explores the polyhedral structure of the QAP-polytope, revealing new facets and inequalities that deepen understanding of its geometry and potential optimization strategies.

Contribution

It identifies a large set of new facets of the QAP-polytope and introduces a general inequality encompassing all known facets, advancing polyhedral combinatorics.

Findings

Discovered exponentially many new facets of the QAP-polytope.

Proposed a unifying inequality for all known and new facets.

Indicated the existence of additional undiscovered facets.

Abstract

The Birkhoff polytope is defined to be the convex hull of permutation matrices, $P_{\sigma}\ \forall \sigma\in S_n$. We define a second-order permutation matrix $P^{[2]}_{\sigma}$ in $\mathbb{R}^{n^2\times n^2}$ corresponding to a permutation $\sigma$ as $(P^{[2]}_{\sigma})_{ij,kl} = (P_{\sigma})_{ij}(P_{\sigma})_{kl}$. We call the convex hull of the second-order permutation matrices, the {\em second-order Birkhoff polytope} and denote it by ${\cal B}^{[2]}$. It can be seen that ${\cal B}^{[2]}$ is isomorphic to the QAP-polytope, the domain of optimization in {\em quadratic assignment problem}. In this work we revisit the polyhedral combinatorics of the QAP-polytope viewing it as ${\cal B}^{[2]}$. Our main contribution is the identification of an exponentially large set of new facets of this polytope. Also we present a general inequality of which all the known facets of this polytope as well as the new ones, that we present in this paper, are special instances. We also establish the existence of more facets which are yet to be identified.