19 research outputs found
Many 2-level polytopes from matroids
The family of 2-level matroids, that is, matroids whose base polytope is
2-level, has been recently studied and characterized by means of combinatorial
properties. 2-level matroids generalize series-parallel graphs, which have been
already successfully analyzed from the enumerative perspective.
We bring to light some structural properties of 2-level matroids and exploit
them for enumerative purposes. Moreover, the counting results are used to show
that the number of combinatorially non-equivalent (n-1)-dimensional 2-level
polytopes is bounded from below by , where
and .Comment: revised version, 19 pages, 7 figure
On vertices and facets of combinatorial 2-level polytopes
2-level polytopes naturally appear in several areas of mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. We investigate upper bounds on the product of the number of facets and the number of vertices, where d is the dimension of a 2-level polytope P. This question was first posed in [3], where experimental results showed an upper bound of d2^{d+1} up to d = 6, where d is the dimension of the polytope. We show that this bound holds for all known (to the best of our knowledge) 2-level polytopes coming from combinatorial settings, including stable set polytopes of perfect graphs and all 2-level base polytopes of matroids. For the latter family, we also give a simple description of the facet-defining inequalities. These results are achieved by an investigation of related combinatorial objects, that could be of independent interest
Recognizing Cartesian products of matrices and polytopes
The 1-product of matrices and is the matrix in whose columns are the concatenation of each column of with
each column of . Our main result is a polynomial time algorithm for the
following problem: given a matrix , is a 1-product, up to permutation of
rows and columns? Our main motivation is a close link between the 1-product of
matrices and the Cartesian product of polytopes, which goes through the concept
of slack matrix. Determining whether a given matrix is a slack matrix is an
intriguing problem whose complexity is unknown, and our algorithm reduces the
problem to irreducible instances. Our algorithm is based on minimizing a
symmetric submodular function that expresses mutual information in information
theory. We also give a polynomial time algorithm to recognize a more
complicated matrix product, called the 2-product. Finally, as a corollary of
our 1-product and 2-product recognition algorithms, we obtain a polynomial time
algorithm to recognize slack matrices of -level matroid base polytopes
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