80 research outputs found
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
Smaller Extended Formulations for the Spanning Tree Polytope of Bounded-genus Graphs
We give an -size extended formulation
for the spanning tree polytope of an -vertex graph embedded on a surface of
genus , improving on the known -size extended formulations
following from Wong and Martin.Comment: v3: fixed some typo
Extension complexity of stable set polytopes of bipartite graphs
The extension complexity of a polytope is the minimum
number of facets of a polytope that affinely projects to . Let be a
bipartite graph with vertices, edges, and no isolated vertices. Let
be the convex hull of the stable sets of . It is easy to
see that . We improve
both of these bounds. For the upper bound, we show that is , which is an improvement when
has quadratically many edges. For the lower bound, we prove that
is when is the
incidence graph of a finite projective plane. We also provide examples of
-regular bipartite graphs such that the edge vs stable set matrix of
has a fooling set of size .Comment: 13 pages, 2 figure
On polyhedral approximations of the positive semidefinite cone
Let be the set of positive semidefinite matrices of trace
equal to one, also known as the set of density matrices. We prove two results
on the hardness of approximating with polytopes. First, we show that if and is an arbitrary matrix of trace equal to one, any
polytope such that must have
linear programming extension complexity at least where is a constant that depends on . Second, we show that any polytope
such that and such that the Gaussian width of is at most
twice the Gaussian width of must have extension complexity at least
. The main ingredient of our proofs is hypercontractivity of
the noise operator on the hypercube.Comment: 12 page
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