A Polyhedral Perspective on the Perfect Matching Lattice

TL;DR

This paper presents a polynomial-time algorithm to construct a basis for the perfect matching lattice of a graph using polyhedral methods, revealing new links between combinatorial and geometric properties of perfect matchings.

Contribution

It introduces a novel polyhedral approach to construct lattice bases of perfect matchings, extending prior theoretical results with a constructive algorithm.

Findings

Provides a polynomial-time algorithm for lattice basis construction

Decomposes graphs into Birkhoff von Neumann subgraphs

Establishes new connections between combinatorial and geometric properties

Abstract

We study the perfect matching lattice of a matching covered graph $G$, generated by the incidence vectors of its perfect matchings. Building on results of Lov\'asz and de Carvalho, Lucchesi, and Murty, we give a polynomial-time algorithm based on polyhedral methods that constructs a lattice basis for this lattice consisting of perfect matchings of $G$. By decomposing along certain odd cuts, we reduce the graph into subgraphs whose perfect matching polytopes coincide with their bipartite relaxations (known as \emph{Birkhoff von Neumann graphs}). This yields a constructive polyhedral proof of the existence of such bases and highlights new connections between combinatorial and geometric properties of perfect matchings.