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**1 - 1**of**1**### Courcelle’s Theorem: An Extension Complexity Analogue∗

, 2015

"... Courcelle’s theorem states that given an MSO formula ϕ and a graph G with n vertices and treewidth τ, checking whether G satisfies ϕ or not can be done in time f(τ, |ϕ|) · n where f is some computable function. We show an analogous result for extension complexity. In particular, we consider the pol ..."

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Courcelle’s theorem states that given an MSO formula ϕ and a graph G with n vertices and treewidth τ, checking whether G satisfies ϕ or not can be done in time f(τ, |ϕ|) · n where f is some computable function. We show an analogous result for extension complexity. In particular, we consider the polytope Pϕ(G) of all satisfying assignments of a given MSO formula ϕ on a given graph G and show that Pϕ(G) can be described by a linear program with f(|ϕ|, τ) · n inequalities where f is some computable function, n is the number of vertices in G and τ is the treewidth of G. In other words, we prove that the extension complexity of Pϕ(G) is linear in the size of the graph G. This provides a first meta theorem about the extension complexity of polytopes related to a wide class of problems and graphs. Furthermore, even though linear time optimization versions of Courcelle’s theorem are known, our result provides a linear size LP for these problems out of the box. We also introduce a simple tool for polyhedral manipulation, called the glued product of polytopes which is a slight generalization of the usual product of polytopes. We use it to build our extended formulation by identifying a case for 0/1 polytopes when the glued product does not increase the extension complexity too much. The glued product may be of independent interest.