Subgraph Polytopes and Independence Polytopes of Count Matroids

TL;DR

This paper explores the relationships between subgraph polytopes, spanning forest polytopes, and independence polytopes of count matroids, providing new extended formulations and bounds in combinatorial optimization.

Contribution

It establishes a strong connection between subgraph polytopes and spanning forest polytopes, and introduces polynomial size extended formulations for independence polytopes of count matroids.

Findings

Polynomial size extended formulations for independence polytopes of count matroids.

New lower bounds on extension complexity of spanning forest polytopes.

Generalization of results to sparsity matroids.

Abstract

Given an undirected graph, the non-empty subgraph polytope is the convex hull of the characteristic vectors of pairs (F, S) where S is a non-empty subset of nodes and F is a subset of the edges with both endnodes in S. We obtain a strong relationship between the non-empty subgraph polytope and the spanning forest polytope. We further show that these polytopes provide polynomial size extended formulations for independence polytopes of count matroids, which generalizes recent results obtained by Iwata et al. referring to sparsity matroids. As a byproduct, we obtain new lower bounds on the extension complexity of the spanning forest polytope in terms of extension complexities of independence polytopes of these matroids.