28 research outputs found
Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix
The positive semidefinite rank of a nonnegative -matrix~ is
the minimum number~ such that there exist positive semidefinite -matrices , such that S(k,\ell) =
\mbox{tr}(A_k^* B_\ell).
The most important, lower bound technique for nonnegative rank is solely
based on the support of the matrix S, i.e., its zero/non-zero pattern. In this
paper, we characterize the power of lower bounds on positive semidefinite rank
based on solely on the support.Comment: 9 page
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
Fooling sets and rank
An matrix is called a \textit{fooling-set matrix of size }
if its diagonal entries are nonzero and for every
. Dietzfelbinger, Hromkovi{\v{c}}, and Schnitger (1996) showed that
n \le (\mbox{rk} M)^2, regardless of over which field the rank is computed,
and asked whether the exponent on \mbox{rk} M can be improved.
We settle this question. In characteristic zero, we construct an infinite
family of rational fooling-set matrices with size n = \binom{\mbox{rk}
M+1}{2}. In nonzero characteristic, we construct an infinite family of
matrices with n= (1+o(1))(\mbox{rk} M)^2.Comment: 10 pages. Now resolves the open problem also in characteristic
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
Smallest Compact Formulation for the Permutahedron
In this note, we consider the permutahedron, the convex hull of all permutations of {1,2…,n} . We show how to obtain an extended formulation for this polytope from any sorting network. By using the optimal Ajtai–Komlós–Szemerédi sorting network, this extended formulation has Θ(nlogn) variables and inequalities. Furthermore, from basic polyhedral arguments, we show that this is best possible (up to a multiplicative constant) since any extended formulation has at least Ω(nlogn) inequalities. The results easily extend to the generalized permutahedron.National Science Foundation (U.S.) (Contract CCF-0829878)National Science Foundation (U.S.) (Contract CCF-1115849)United States. Office of Naval Research (Grant 0014-05-1-0148