Complete Description of Matching Polytopes with One Linearized Quadratic Term for Bipartite Graphs

TL;DR

This paper fully characterizes the convex hulls of matchings in complete bipartite graphs extended by a binary variable for a specific edge pair, providing a complete inequality description and extending to capacitated b-matchings.

Contribution

It provides the first complete irredundant inequality description for these polytopes, settling a conjecture and deriving new facetness and separation results.

Findings

Complete inequality description of the polytopes.

Resolution of Klein's conjecture from 2015.

Extension of results to capacitated b-matchings.

Abstract

We consider, for complete bipartite graphs, the convex hulls of characteristic vectors of all matchings, extended by a binary entry indicating whether the matching contains two specific edges. These polytopes are associated to the quadratic matching problems with a single linearized quadratic term. We provide a complete irredundant inequality description, which settles a conjecture by Klein (Ph.D. thesis, TU Dortmund, 2015). In addition, we also derive facetness and separation results for the polytopes. The completeness proof is based on a geometric relationship to a matching polytope of a nonbipartite graph. Using standard techniques, we finally extend the result to capacitated b-matchings.