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

Abstract

Rothvoss [1] showed 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 2Ω(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 2Ω(n) and an affine space. Our proof relies on a new technique to rescale semidefinite factorizations. © 2013 Springer-Verlag.