16,653 research outputs found
Random curves on surfaces induced from the Laplacian determinant
We define natural probability measures on cycle-rooted spanning forests
(CRSFs) on graphs embedded on a surface with a Riemannian metric. These
measures arise from the Laplacian determinant and depend on the choice of a
unitary connection on the tangent bundle to the surface.
We show that, for a sequence of graphs conformally approximating the
surface, the measures on CRSFs of converge and give a limiting
probability measure on finite multicurves (finite collections of pairwise
disjoint simple closed curves) on the surface, independent of the approximating
sequence.
Wilson's algorithm for generating spanning trees on a graph generalizes to a
cycle-popping algorithm for generating CRSFs for a general family of weights on
the cycles. We use this to sample the above measures. The sampling algorithm,
which relates these measures to the loop-erased random walk, is also used to
prove tightness of the sequence of measures, a key step in the proof of their
convergence.
We set the framework for the study of these probability measures and their
scaling limits and state some of their properties
Distributed Approximation of Maximum Independent Set and Maximum Matching
We present a simple distributed -approximation algorithm for maximum
weight independent set (MaxIS) in the model which completes
in rounds, where is the maximum
degree, is the number of rounds needed to compute a maximal
independent set (MIS) on , and is the maximum weight of a node. %Whether
our algorithm is randomized or deterministic depends on the \texttt{MIS}
algorithm used as a black-box.
Plugging in the best known algorithm for MIS gives a randomized solution in
rounds, where is the number of nodes.
We also present a deterministic -round algorithm based
on coloring.
We then show how to use our MaxIS approximation algorithms to compute a
-approximation for maximum weight matching without incurring any additional
round penalty in the model. We use a known reduction for
simulating algorithms on the line graph while incurring congestion, but we show
our algorithm is part of a broad family of \emph{local aggregation algorithms}
for which we describe a mechanism that allows the simulation to run in the
model without an additional overhead.
Next, we show that for maximum weight matching, relaxing the approximation
factor to () allows us to devise a distributed algorithm
requiring rounds for any constant
. For the unweighted case, we can even obtain a
-approximation in this number of rounds. These algorithms are
the first to achieve the provably optimal round complexity with respect to
dependency on
Constant-Factor Approximation for TSP with Disks
We revisit the traveling salesman problem with neighborhoods (TSPN) and
present the first constant-ratio approximation for disks in the plane: Given a
set of disks in the plane, a TSP tour whose length is at most times
the optimal can be computed in time that is polynomial in . Our result is
the first constant-ratio approximation for a class of planar convex bodies of
arbitrary size and arbitrary intersections. In order to achieve a
-approximation, we reduce the traveling salesman problem with disks, up
to constant factors, to a minimum weight hitting set problem in a geometric
hypergraph. The connection between TSPN and hitting sets in geometric
hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure
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