Bounds on the number of 2-level polytopes, cones and configurations

06/15/2018

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by   Samuel Fiorini, et al.

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We prove an upper bound of the form 2^O(d^2 polylog d) on the number of affine (resp. linear) equivalence classes of, by increasing order of generality, 2-level d-polytopes, d-cones and d-configurations. This in particular answers positively a conjecture of Bohn et al. on 2-level polytopes. We obtain our upper bound by relating affine (resp. linear) equivalence classes of 2-level d-polytopes, d-cones and d-configurations to faces of the correlation cone. We complement this with a 2^Ω(d^2) lower bound, by estimating the number of nonequivalent stable set polytopes of bipartite graphs.