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

Jop Briët; Daniel Dadush; Sebastian Pokutta

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arXiv · 1305.3268

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

Jop Briët·Daniel Dadush·Sebastian Pokutta

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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.

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BibTeX RIS

Jop Briët, Daniel Dadush, Sebastian Pokutta. 2013-05-14. On the existence of 0/1 polytopes with high semidefinite extension complexity. https://arxiv.org/abs/1305.3268

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