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

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).

v2: Minor changes following the referees' comments

References in corpus (1)

• Matroid polytopes, nested sets and Bergman fans

Cited by in corpus (1)

• Graphs with nonnegative resistance curvature