22 research outputs found
On the Linear Extension Complexity of Regular n-gons
In this paper, we propose new lower and upper bounds on the linear extension
complexity of regular -gons. Our bounds are based on the equivalence between
the computation of (i) an extended formulation of size of a polytope ,
and (ii) a rank- nonnegative factorization of a slack matrix of the polytope
. The lower bound is based on an improved bound for the rectangle covering
number (also known as the boolean rank) of the slack matrix of the -gons.
The upper bound is a slight improvement of the result of Fiorini, Rothvoss and
Tiwary [Extended Formulations for Polygons, Discrete Comput. Geom. 48(3), pp.
658-668, 2012]. The difference with their result is twofold: (i) our proof uses
a purely algebraic argument while Fiorini et al. used a geometric argument, and
(ii) we improve the base case allowing us to reduce their upper bound by one when for some integer . We conjecture that this new upper bound
is tight, which is suggested by numerical experiments for small . Moreover,
this improved upper bound allows us to close the gap with the best known lower
bound for certain regular -gons (namely, and ) hence allowing for the first time to determine their extension
complexity.Comment: 20 pages, 3 figures. New contribution: improved lower bound for the
boolean rank of the slack matrices of n-gon
Small Extended Formulations for Cyclic Polytopes
We provide an extended formulation of size O(log n)^{\lfloor d/2 \rfloor} for
the cyclic polytope with dimension d and n vertices (i,i^2,\ldots,i^d), i in
[n]. First, we find an extended formulation of size log(n) for d= 2. Then, we
use this as base case to construct small-rank nonnegative factorizations of the
slack matrices of higher-dimensional cyclic polytopes, by iterated tensor
products. Through Yannakakis's factorization theorem, these factorizations
yield small-size extended formulations for cyclic polytopes of dimension d>2
An upper bound for nonnegative rank
We provide a nontrivial upper bound for the nonnegative rank of rank-three
matrices, which allows us to prove that [6(n+1)/7] linear inequalities suffice
to describe a convex n-gon up to a linear projection
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
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
Polytopes of Minimum Positive Semidefinite Rank
The positive semidefinite (psd) rank of a polytope is the smallest for
which the cone of real symmetric psd matrices admits an affine
slice that projects onto the polytope. In this paper we show that the psd rank
of a polytope is at least the dimension of the polytope plus one, and we
characterize those polytopes whose psd rank equals this lower bound. We give
several classes of polytopes that achieve the minimum possible psd rank
including a complete characterization in dimensions two and three
Algorithms for Positive Semidefinite Factorization
This paper considers the problem of positive semidefinite factorization (PSD
factorization), a generalization of exact nonnegative matrix factorization.
Given an -by- nonnegative matrix and an integer , the PSD
factorization problem consists in finding, if possible, symmetric -by-
positive semidefinite matrices and such
that for , and . PSD
factorization is NP-hard. In this work, we introduce several local optimization
schemes to tackle this problem: a fast projected gradient method and two
algorithms based on the coordinate descent framework. The main application of
PSD factorization is the computation of semidefinite extensions, that is, the
representations of polyhedrons as projections of spectrahedra, for which the
matrix to be factorized is the slack matrix of the polyhedron. We compare the
performance of our algorithms on this class of problems. In particular, we
compute the PSD extensions of size for the
regular -gons when , and . We also show how to generalize our
algorithms to compute the square root rank (which is the size of the factors in
a PSD factorization where all factor matrices and have rank one)
and completely PSD factorizations (which is the special case where the input
matrix is symmetric and equality is required for all ).Comment: 21 pages, 3 figures, 3 table