A Short Proof that the Extension Complexity of the Correlation Polytope Grows Exponentially

TL;DR

This paper provides a concise proof demonstrating that the extension complexity of the correlation polytope grows exponentially with dimension, specifically at least 1.5^n, improving previous bounds.

Contribution

It introduces a simple combinatorial argument to establish exponential lower bounds on the extension complexity of the correlation polytope, enhancing prior results.

Findings

Extension complexity of correlation polytope is at least 1.5^n.

Rectangle covering number of the disjointness matrix is at least 1.5^n.

Improves previous lower bounds on extension complexity and communication complexity.

Abstract

We establish that the extension complexity of the nXn correlation polytope is at least 1.5^n by a short proof that is self-contained except for using the fact that every face of a polyhedron is the intersection of all facets it is contained in. The main innovative aspect of the proof is a simple combinatorial argument showing that the rectangle covering number of the unique-disjointness matrix is at least 1.5^n, and thus the nondeterministic communication complexity of the unique-disjointness predicate is at least .58n. We thereby slightly improve on the previously best known lower bounds 1.24^n and .31n, respectively.