On the extension complexity of polytopes separating subsets of the Boolean cube

05/25/2021

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by   Pavel Hrubes, et al.

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We show that 1. for every A⊆{0, 1}^n, there exists a polytope P⊆ℝ^n with P ∩{0, 1}^n = A and extension complexity O(2^n/2), 2. there exists an A⊆{0, 1}^n such that the extension complexity of any P with P∩{0, 1}^n = A must be at least 2^n/3(1-o(1)). We also remark that the extension complexity of any 0/1-polytope in ℝ^n is at most O(2^n/n) and pose the problem whether the upper bound can be improved to O(2^cn), for c<1.