The Exact Bipartite Matching Polytope Has Exponential Extension Complexity

TL;DR

This paper proves that the convex hull of certain perfect matchings in bipartite graphs cannot be represented by small linear programs, highlighting fundamental complexity limitations in combinatorial optimization.

Contribution

It establishes exponential lower bounds on the size of linear programs for describing specific perfect matching polytopes in bipartite graphs.

Findings

No sub-exponential linear program describes the convex hull of exact matchings.

Stronger result: no small LP describes the convex hull of perfect matchings with an odd number of red edges.

Highlights inherent complexity in linear programming formulations for combinatorial problems.

Abstract

Given a graph with edges colored red or blue and an integer $k$, the exact perfect matching problem asks if there exists a perfect matching with exactly $k$ red edges. There exists a randomized polylogarithmic-time parallel algorithm to solve this problem, dating back to the eighties, but no deterministic polynomial-time algorithm is known, even for bipartite graphs. In this paper we show that there is no sub-exponential sized linear program that can describe the convex hull of exact matchings in bipartite graphs. In fact, we prove something stronger, that there is no sub-exponential sized linear program to describe the convex hull of perfect matchings with an odd number of red edges.