Extended Formulations for Sparsity Matroids

TL;DR

This paper presents a polynomial-size extended formulation for the base polytope of $(k,\, ext{ell})$-sparsity matroids, improving understanding of their polyhedral structure and employing advanced communication protocol techniques.

Contribution

It introduces the first polynomial-size extended formulation for these sparsity matroids' base polytopes, utilizing a novel randomized communication protocol approach.

Findings

Formulation size is $O(|V||E|)$ for $k \,\geq\, \ell$.

Formulation size is $O(|V|^2 |E|)$ for $k \,\leq\, \ell$.

Employs a recent technique by Faenza et al. for the construction.

Abstract

We show the existence of a polynomial-size extended formulation for the base polytope of a $(k,\ell)$-sparsity matroid. For an undirected graph $G=(V,E)$, the size of the formulation is $O(|V||E|)$ when $k \geq \ell$ and $O(|V|^2 |E|)$ when $k \leq \ell$. To this end, we employ the technique developed by Faenza et al. recently that uses a randomized communication protocol.