15 research outputs found
On the existence of 0/1 polytopes with high semidefinite extension complexity
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
On the existence of 0/1 polytopes with high semidefinite extension complexity
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 2Ω(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
2Ω(n) and an affine space. Our proof relies on a new technique to rescale semidefinite
factorizations
Computing approximate PSD factorizations
We give an algorithm for computing approximate PSD factorizations of
nonnegative matrices. The running time of the algorithm is polynomial in the
dimensions of the input matrix, but exponential in the PSD rank and the
approximation error. The main ingredient is an exact factorization algorithm
when the rows and columns of the factors are constrained to lie in a general
polyhedron. This strictly generalizes nonnegative matrix factorizations which
can be captured by letting this polyhedron to be the nonnegative orthant.Comment: 10 page
The matching polytope does not admit fully-polynomial size relaxation schemes
The groundbreaking work of Rothvo{\ss} [arxiv:1311.2369] established that
every linear program expressing the matching polytope has an exponential number
of inequalities (formally, the matching polytope has exponential extension
complexity). We generalize this result by deriving strong bounds on the
polyhedral inapproximability of the matching polytope: for fixed , every polyhedral -approximation
requires an exponential number of inequalities, where is the number of
vertices. This is sharp given the well-known -approximation of size
provided by the odd-sets of size up to
. Thus matching is the first problem in , whose natural
linear encoding does not admit a fully polynomial-size relaxation scheme (the
polyhedral equivalent of an FPTAS), which provides a sharp separation from the
polynomial-size relaxation scheme obtained e.g., via constant-sized odd-sets
mentioned above.
Our approach reuses ideas from Rothvo{\ss} [arxiv:1311.2369], however the
main lower bounding technique is different. While the original proof is based
on the hyperplane separation bound (also called the rectangle corruption
bound), we employ the information-theoretic notion of common information as
introduced in Braun and Pokutta [http://eccc.hpi-web.de/report/2013/056/],
which allows to analyze perturbations of slack matrices. It turns out that the
high extension complexity for the matching polytope stem from the same source
of hardness as for the correlation polytope: a direct sum structure.Comment: 21 pages, 3 figure
Semidefinite descriptions of the convex hull of rotation matrices
We study the convex hull of , thought of as the set of
orthogonal matrices with unit determinant, from the point of view of
semidefinite programming. We show that the convex hull of is doubly
spectrahedral, i.e. both it and its polar have a description as the
intersection of a cone of positive semidefinite matrices with an affine
subspace. Our spectrahedral representations are explicit, and are of minimum
size, in the sense that there are no smaller spectrahedral representations of
these convex bodies.Comment: 29 pages, 1 figur
Exponential Lower Bounds for Polytopes in Combinatorial Optimization
We solve a 20-year old problem posed by Yannakakis and prove that there
exists no polynomial-size linear program (LP) whose associated polytope
projects to the traveling salesman polytope, even if the LP is not required to
be symmetric. Moreover, we prove that this holds also for the cut polytope and
the stable set polytope. These results were discovered through a new connection
that we make between one-way quantum communication protocols and semidefinite
programming reformulations of LPs.Comment: 19 pages, 4 figures. This version of the paper will appear in the
Journal of the ACM. The earlier conference version in STOC'12 had the title
"Linear vs. Semidefinite Extended Formulations: Exponential Separation and
Strong Lower Bounds