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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 treepolytope of $G$ is $O(n^{3/2})$. This improves on the $O(n^2)$ bounds followingfrom the work of Wong (1980) and Martin (1991). It also extends a result ofFiorini, Huynh, Joret, and Pashkovich (2017), who obtained a $O(n^{3/2})$ boundfor graphs embedded in a fixed surface. Our proof works more generally for allgraph classes admitting strongly sublinear balanced separators: We prove thatfor every constant $\beta$ with $0<\beta<1$, if $\mathcal G$ is a graph classclosed under induced subgraphs such that all $n$-vertex graphs in $\mathcal G$have balanced separators of size $O(n^\beta)$, then the extension complexity ofthe 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 directconstruction of the extended formulation, the other is via communicationprotocols. Using the latter approach we also give a short proof of the $O(n)$bound for planar graphs due to Williams (2002).

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