9,517 research outputs found
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
On (2,3)-agreeable Box Societies
The notion of -agreeable society was introduced by Deborah Berg et
al.: a family of convex subsets of is called -agreeable if any
subfamily of size contains at least one non-empty -fold intersection. In
that paper, the -agreeability of a convex family was shown to imply the
existence of a subfamily of size with non-empty intersection, where
is the size of the original family and is an explicit
constant depending only on and . The quantity is called
the minimal \emph{agreement proportion} for a -agreeable family in
.
If we only assume that the sets are convex, simple examples show that
for -agreeable families in where . In this paper,
we introduce new techniques to find positive lower bounds when restricting our
attention to families of -boxes, i.e. cuboids with sides parallel to the
coordinates hyperplanes. We derive explicit formulas for the first non-trivial
case: the case of -agreeable families of -boxes with .Comment: 15 pages, 10 figure
An update on the Hirsch conjecture
The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to
George Dantzig. It states that the graph of a d-dimensional polytope with n
facets cannot have diameter greater than n - d.
Despite being one of the most fundamental, basic and old problems in polytope
theory, what we know is quite scarce. Most notably, no polynomial upper bound
is known for the diameters that are conjectured to be linear. In contrast, very
few polytopes are known where the bound is attained. This paper collects
known results and remarks both on the positive and on the negative side of the
conjecture. Some proofs are included, but only those that we hope are
accessible to a general mathematical audience without introducing too many
technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2
and put into the appendix arXiv:0912.423
Faces of Birkhoff Polytopes
The Birkhoff polytope B(n) is the convex hull of all (n x n) permutation
matrices, i.e., matrices where precisely one entry in each row and column is
one, and zeros at all other places. This is a widely studied polytope with
various applications throughout mathematics.
In this paper we study combinatorial types L of faces of a Birkhoff polytope.
The Birkhoff dimension bd(L) of L is the smallest n such that B(n) has a face
with combinatorial type L.
By a result of Billera and Sarangarajan, a combinatorial type L of a
d-dimensional face appears in some B(k) for k less or equal to 2d, so bd(L) is
at most d. We will characterize those types whose Birkhoff dimension is at
least 2d-3, and we prove that any type whose Birkhoff dimension is at least d
is either a product or a wedge over some lower dimensional face. Further, we
computationally classify all d-dimensional combinatorial types for d between 2
and 8.Comment: 29 page
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