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On the existence of 0/1 polytopes with high semidefinite extension complexity. (English) Zbl 1395.90241
Bodlaender, Hans L. (ed.) et al., Algorithms – ESA 2013. 21st annual European symposium, Sophia Antipolis, France, September 2–4, 2013. Proceedings. Berlin: Springer (ISBN 978-3-642-40449-8/pbk). Lecture Notes in Computer Science 8125, 217-228 (2013).
Summary: T. Rothvoß [Math. Program. 142, No. 1–2 (A), 255–268 (2013; Zbl 1282.90245)] 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 \(2^{\Omega (n)}\) 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.
For the entire collection see [Zbl 1270.68017].

MSC:
90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
68Q25 Analysis of algorithms and problem complexity
90C22 Semidefinite programming
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