On the Extension Complexity of Polytopes Separating Subsets of the Boolean Cube

Abstract

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