Extended formulations for matroid polytopes through randomized protocols

TL;DR

This paper demonstrates that base polytopes of matroids have extended formulations whose size is linearly related to their hitting number, utilizing a connection between extended formulations and communication protocols.

Contribution

It introduces a new method to construct extended formulations for matroid polytopes based on randomized protocols, generalizing previous results for spanning tree polytopes.

Findings

Extended formulations for matroid polytopes depend linearly on hitting number.

The approach simplifies previous constructions and broadens their applicability.

Connects extended formulations with communication protocols for polyhedral descriptions.

Abstract

Let $P$ be a polytope. The hitting number of $P$ is the smallest size of a hitting set of the facets of $P$, i.e., a subset of vertices of $P$ such that every facet of $P$ has a vertex in the subset. An extended formulation of $P$ is the description of a polyhedron that linearly projects to $P$. We show that, if $P$ is the base polytope of any matroid, then $P$ admits an extended formulation whose size depends linearly on the hitting number of $P$. Our extended formulations generalize those of the spanning tree polytope given by Martin and Wong. Our proof is simple and short, and it goes through the deep connection between extended formulations and communication protocols.