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

TL;DR

This paper investigates the extension complexity of polytopes that separate subsets of the Boolean cube, establishing upper bounds for all subsets and lower bounds for some, highlighting gaps and open problems in polytope complexity.

Contribution

It provides the first bounds on extension complexity for arbitrary subset-separating polytopes and identifies significant gaps between these bounds.

Findings

Existence of polytopes with extension complexity O(2^{n/2}) for any subset

Existence of subsets requiring extension complexity at least 2^{n/3(1-o(1))}

Extension complexity of 0/1-polytopes is at most O(2^n/n)

Abstract

We show that 1. 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 $O(2^{n/2})$, 2. 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 $2^{\frac{n}{3}(1-o(1))}$. 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 $c<1$.