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

Abstract

In Rothvoß (Math Program 142(1-2):255-268, 2013) 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 Ω(n) facets, i.e., its linear extension complexity is exponential. The question whether there exists a 0/1 polytope with high positive semidefinite 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 Ω(n) and an affine space. Our proof relies on a new technique to rescale semidefinite factorizations.

Maximum semidefinite and linear extension complexity of families of polytopes

“…While the approaches in [21,8,4] can be applied to further families of polytopes, it seems that, dealing with a new family, one is forced to repeat large parts of the argumentation in the above sources. In contrast, in this paper, we present a theorem which can be used as a simple tool for finding lower bounds on the maximum semidefinite and linear extension complexity for general families of polytopes.…”

“…Besides the simple applicability of Theorem 1, we view its short and simple proof as an essential contribution. Note that the original proofs in [21,8,4] turn out to be quite long and require a number of non-trivial tools. They rely on a counting argument developed in [21] based on encoding extended formulations by certain kinds of factorizations of slack matrices of polytopes (see [26,14,10]) whose components have to be carefully balanced and rounded.…”

“…We further show how our result can be used to improve on the corresponding bounds known for polygons with integer vertices.Our geometric proof builds upon nothing else than a simple well-known property of maximum volume inscribed ellipsoids of convex bodies. In particular, it does not rely on factorizations over the semidefinite cone and thus avoids involved procedures of balancing them as required, e.g., in [4]. We hope that revealing the geometry behind the phenomenon opens doors for further results.Moreover, we show that the linear extension complexity of every d-dimensional 0/1-polytope is bounded from above by O( 2 d d ).…”

“…Our geometric proof builds upon nothing else than a simple well-known property of maximum volume inscribed ellipsoids of convex bodies. In particular, it does not rely on factorizations over the semidefinite cone and thus avoids involved procedures of balancing them as required, e.g., in [4]. We hope that revealing the geometry behind the phenomenon opens doors for further results.…”

The matching problem has no small symmetric SDP

“…Some recent work has shown limits to the power of small SDPs. Briët et al [2013Briët et al [ , 2015 nonconstructively give an exponential lower bound on the size of SDP formulations for most 0/1 polytopes. Lee et al [2015] give an exponential lower bound for solving the traveling salesperson problem (TSP) and approximating max 3-sat.…”