A (convex) polytope P is said to be 2-level if for every direction of hyperplanes which is facet-defining for P , the vertices of P can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of 2-level polytopes of a given dimension d, and provide complete experimental results for d 7. Our approach is inductive: for each fixed (d − 1)-dimensional 2-level polytope P 0 , we enumerate all d-dimensional 2-level polytopes P that have P 0 as a facet. This relies on the enumeration of the closed sets of a closure operator over a finite ground set. By varying the prescribed facet P 0 , we obtain all 2-level polytopes in dimension d.(1,3,4,5) Université libre de Bruxelles, Brussels, Belgium Lemma 4. Let P be a d-polytope having facet-defining inequalities g 1 (x) 0, . . . , g m (x) 0, and vertices v 1 , . . . , v n . If σ denotes a map from the affine hull aff(P ) of P to R m defined by σ(x) i := g i (x) for all x ∈ aff(P ), then the polytopes P and σ(P ) are affinely equivalent.Proof. The map σ : aff(P ) → R m is affine, and injective because it maps the vertices of any simplicial core for P to affinely independent points. The result follows.By definition, a polytope P is 2-level if and only if S(P ) can be scaled to be a 0/1 matrix. Given a 2-level polytope, we henceforth always assume that its facet-defining inequalities are scaled so that the slacks are 0/1. Thus, the slack embedding of a 2-level polytope depends only on the support 1 of its slack matrix, which only depends on its combinatorial structure. As a consequence, we have the following result:Lemma 5. Two 2-level polytopes are affinely equivalent if and only if they have the same combinatorial type.

Abstract.The extension complexity xc(P ) of a polytope P is the minimum number of facets of a polytope that affinely projects to P . Let G be a bipartite graph with n vertices, m edges, and no isolated vertices. Let STAB(G) be the convex hull of the stable sets of G. It is easy to see that n xc(STAB(G)) n + m. We improve both of these bounds. For the upper bound, we show that xc(STAB(G)) is O( n 2 log n ), which is an improvement when G has quadratically many edges. For the lower bound, we prove that xc(STAB(G)) is Ω(n log n) when G is the incidence graph of a finite projective plane. We also provide examples of 3-regular bipartite graphs G such that the edge vs stable set matrix of G has a fooling set of size |E(G)|.

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We prove that for every n-vertex graph G, the extension complexity of the correlation polytope of G is 2 O(tw(G)+log n) , where tw(G) is the treewidth of G. Our main result is that this bound is tight for graphs contained in minor-closed classes.

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