4,981 research outputs found
A QPTAS for the Base of the Number of Triangulations of a Planar Point Set
The number of triangulations of a planar n point set is known to be ,
where the base lies between and The fastest known algorithm
for counting triangulations of a planar n point set runs in time.
The fastest known arbitrarily close approximation algorithm for the base of the
number of triangulations of a planar n point set runs in time subexponential in
We present the first quasi-polynomial approximation scheme for the base of
the number of triangulations of a planar point set
Trisections of a 3-rotationally symmetric planar convex body minimizing the maximum relative diameter
In this work we study the fencing problem consisting of finnding a trisection
of a 3-rotationally symmetric planar convex body which minimizes the maximum
relative diameter. We prove that an optimal solution is given by the so-called
standard trisection. We also determine the optimal set giving the minimum value
for this functional and study the corresponding universal lower bound.Comment: Preliminary version, 20 pages, 15 figure
Approximation Schemes for Partitioning: Convex Decomposition and Surface Approximation
We revisit two NP-hard geometric partitioning problems - convex decomposition
and surface approximation. Building on recent developments in geometric
separators, we present quasi-polynomial time algorithms for these problems with
improved approximation guarantees.Comment: 21 pages, 6 figure
Kasteleyn cokernels
We consider Kasteleyn and Kasteleyn-Percus matrices, which arise in
enumerating matchings of planar graphs, up to matrix operations on their rows
and columns. If such a matrix is defined over a principal ideal domain, this is
equivalent to considering its Smith normal form or its cokernel. Many
variations of the enumeration methods result in equivalent matrices. In
particular, Gessel-Viennot matrices are equivalent to Kasteleyn-Percus
matrices.
We apply these ideas to plane partitions and related planar of tilings. We
list a number of conjectures, supported by experiments in Maple, about the
forms of matrices associated to enumerations of plane partitions and other
lozenge tilings of planar regions and their symmetry classes. We focus on the
case where the enumerations are round or -round, and we conjecture that
cokernels remain round or -round for related ``impossible enumerations'' in
which there are no tilings. Our conjectures provide a new view of the topic of
enumerating symmetry classes of plane partitions and their generalizations. In
particular we conjecture that a -specialization of a Jacobi-Trudi matrix has
a Smith normal form. If so it could be an interesting structure associated to
the corresponding irreducible representation of \SL(n,\C). Finally we find,
with proof, the normal form of the matrix that appears in the enumeration of
domino tilings of an Aztec diamond.Comment: 14 pages, 19 in-line figures. Very minor copy correction
Generating Random Elements of Finite Distributive Lattices
This survey article describes a method for choosing uniformly at random from
any finite set whose objects can be viewed as constituting a distributive
lattice. The method is based on ideas of the author and David Wilson for using
``coupling from the past'' to remove initialization bias from Monte Carlo
randomization. The article describes several applications to specific kinds of
combinatorial objects such as tilings, constrained lattice paths, and
alternating-sign matrices.Comment: 13 page
Enumeration of Matchings: Problems and Progress
This document is built around a list of thirty-two problems in enumeration of
matchings, the first twenty of which were presented in a lecture at MSRI in the
fall of 1996. I begin with a capsule history of the topic of enumeration of
matchings. The twenty original problems, with commentary, comprise the bulk of
the article. I give an account of the progress that has been made on these
problems as of this writing, and include pointers to both the printed and
on-line literature; roughly half of the original twenty problems were solved by
participants in the MSRI Workshop on Combinatorics, their students, and others,
between 1996 and 1999. The article concludes with a dozen new open problems.
(Note: This article supersedes math.CO/9801060 and math.CO/9801061.)Comment: 1+37 pages; to appear in "New Perspectives in Geometric
Combinatorics" (ed. by Billera, Bjorner, Green, Simeon, and Stanley),
Mathematical Science Research Institute publication #37, Cambridge University
Press, 199
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