Smaller extended formulations for spanning tree polytopes in minor-closed classes and beyond

TL;DR

This paper establishes improved upper bounds on the extension complexity of spanning tree polytopes for graphs in minor-closed classes, generalizing previous results and introducing new proof techniques.

Contribution

It provides a unified framework for bounding extension complexity in graph classes with sublinear balanced separators, extending known results to broader classes.

Findings

Extension complexity of spanning tree polytopes is O(n^{3/2}) for minor-closed classes.

Generalization to graph classes with sublinear balanced separators, achieving O(n^{1+eta}) bounds.

Two proof methods: direct construction and communication protocols, including a new proof for planar graphs.

Abstract

Let $G$ be a connected $n$-vertex graph in a proper minor-closed class $\mathcal G$. We prove that the extension complexity of the spanning tree polytope of $G$ is $O(n^{3/2})$. This improves on the $O(n^2)$ bounds following from the work of Wong (1980) and Martin (1991). It also extends a result of Fiorini, Huynh, Joret, and Pashkovich (2017), who obtained a $O(n^{3/2})$ bound for graphs embedded in a fixed surface. Our proof works more generally for all graph classes admitting strongly sublinear balanced separators: We prove that for every constant $\beta$ with $0<\beta<1$, if $\mathcal G$ is a graph class closed under induced subgraphs such that all $n$-vertex graphs in $\mathcal G$ have balanced separators of size $O(n^\beta)$, then the extension complexity of the spanning tree polytope of every connected $n$-vertex graph in $\mathcal{G}$ is $O(n^{1+\beta})$. We in fact give two proofs of this result, one is a direct construction of the extended formulation, the other is via communication protocols. Using the latter approach we also give a short proof of the $O(n)$ bound for planar graphs due to Williams (2002).