Markovian protocols and an upper bound on the extension complexity of the matching polytope

TL;DR

This paper introduces a geometric framework using Markovian protocols to analyze the extension complexity of polytopes, providing new bounds for the matching polytope and recovering known formulations for the permutahedron.

Contribution

It develops a Markovian protocol-based characterization of extension complexity, leading to improved bounds for the matching polytope and a new proof for the permutahedron's formulation.

Findings

Derived a $ ilde{O}(n^3 ext{·}1.5^n)$ upper bound for the matching polytope's extension complexity.

Reproduced Goemans' compact permutahedron formulation using a one-round protocol.

Provided a geometric perspective linking communication protocols to polytope extension complexity.

Abstract

This paper investigates the extension complexity of polytopes by exploiting the correspondence between non-negative factorizations of slack matrices and randomized communication protocols. We introduce a geometric characterization of extension complexity based on the width of Markovian protocols, as a variant of the framework introduced by Faenza et al. This enables us to derive a new upper bound of $\tilde{O}(n^3\cdot 1.5^n)$ for the extension complexity of the matching polytope $P_{\text{match}}(n)$, improving upon the standard $2^n$-bound given by Edmonds' description. Additionally, we recover Goemans' compact formulation for the permutahedron using a one-round protocol based on sorting networks.