On the existence of 0/1 polytopes with high semidefinite extension complexity

TL;DR

This paper proves the existence of 0/1 polytopes with high semidefinite extension complexity, demonstrating that such polytopes require large spectrahedra to project onto, thus extending known exponential complexity results.

Contribution

It establishes the first known example of 0/1 polytopes with high PSD extension complexity, using a novel rescaling technique for semidefinite factorizations.

Findings

Existence of 0/1 polytopes with exponential PSD extension complexity

Any spectrahedron projecting to such polytopes must be very high-dimensional

Introduces a new rescaling technique for semidefinite factorizations

Abstract

In Rothvo\ss{} it was shown that there exists a 0/1 polytope (a polytope whose vertices are in \{0,1\}^{n}) such that any higher-dimensional polytope projecting to it must have 2^{\Omega(n)} facets, i.e., its linear extension complexity is exponential. The question whether there exists a 0/1 polytope with high PSD extension complexity was left open. We answer this question in the affirmative by showing that there is a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension~2^{\Omega(n)} and an affine space. Our proof relies on a new technique to rescale semidefinite factorizations.