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

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.