12,227 research outputs found
Distributed Dominating Set Approximations beyond Planar Graphs
The Minimum Dominating Set (MDS) problem is one of the most fundamental and
challenging problems in distributed computing. While it is well-known that
minimum dominating sets cannot be approximated locally on general graphs, over
the last years, there has been much progress on computing local approximations
on sparse graphs, and in particular planar graphs.
In this paper we study distributed and deterministic MDS approximation
algorithms for graph classes beyond planar graphs. In particular, we show that
existing approximation bounds for planar graphs can be lifted to bounded genus
graphs, and present (1) a local constant-time, constant-factor MDS
approximation algorithm and (2) a local -time
approximation scheme. Our main technical contribution is a new analysis of a
slightly modified variant of an existing algorithm by Lenzen et al.
Interestingly, unlike existing proofs for planar graphs, our analysis does not
rely on direct topological arguments.Comment: arXiv admin note: substantial text overlap with arXiv:1602.0299
Testing bounded arboricity
In this paper we consider the problem of testing whether a graph has bounded
arboricity. The family of graphs with bounded arboricity includes, among
others, bounded-degree graphs, all minor-closed graph classes (e.g. planar
graphs, graphs with bounded treewidth) and randomly generated preferential
attachment graphs. Graphs with bounded arboricity have been studied extensively
in the past, in particular since for many problems they allow for much more
efficient algorithms and/or better approximation ratios.
We present a tolerant tester in the sparse-graphs model. The sparse-graphs
model allows access to degree queries and neighbor queries, and the distance is
defined with respect to the actual number of edges. More specifically, our
algorithm distinguishes between graphs that are -close to having
arboricity and graphs that -far from having
arboricity , where is an absolute small constant. The query
complexity and running time of the algorithm are
where denotes
the number of vertices and denotes the number of edges. In terms of the
dependence on and this bound is optimal up to poly-logarithmic factors
since queries are necessary (and .
We leave it as an open question whether the dependence on can be
improved from quasi-polynomial to polynomial. Our techniques include an
efficient local simulation for approximating the outcome of a global (almost)
forest-decomposition algorithm as well as a tailored procedure of edge
sampling
Triaxial Galaxies with Cusps
We have constructed fully self-consistent models of triaxial galaxies with
central density cusps. The triaxial generalizations of Dehnen's spherical
models are presented, which have densities that vary as 1/r^gamma near the
center and 1/r^4 at large radii. We computed libraries of about 7000 orbits in
each of two triaxial models with gamma=1 (weak cusp) and gamma=2 (strong cusp);
these two models have density profiles similar to those of the core and
power-law galaxies observed by HST. Both mass models have short-to-long axis
ratios of 1:2 and are maximally triaxial. A large fraction of the orbits in
both model potentials are stochastic, as evidenced by their non-zero Liapunov
exponents. We show that most of the stochastic orbits in the strong- cusp
potential diffuse relatively quickly through their allowed phase-space volumes,
on time scales of 100 - 1000 dynamical times. Stochastic orbits in the
weak-cusp potential diffuse more slowly, often retaining their box-like shapes
for 1000 dynamical times or longer. Attempts to construct self- consistent
solutions using just the regular orbits failed for both mass models.
Quasi-equilibrium solutions that include the stochastic orbits exist for both
models; however, real galaxies constructed in this way would evolve near the
center due to the continued mixing of the stochastic orbits. We attempted to
construct more nearly stationary models in which stochastic phase space was
uniformly populated at low energies. These ``fully mixed'' solutions were found
to exist only for the weak-cusp potential. No significant fraction of the mass
could be placed on fully-mixed stochastic orbits in the strong-cusp model,
demonstrating that strong triaxiality can be inconsistent with a high central
density.Comment: 58 TEX pages, 14 PostScript figures, uses epsf.st
Conjugate Bayes for probit regression via unified skew-normal distributions
Regression models for dichotomous data are ubiquitous in statistics. Besides
being useful for inference on binary responses, these methods serve also as
building blocks in more complex formulations, such as density regression,
nonparametric classification and graphical models. Within the Bayesian
framework, inference proceeds by updating the priors for the coefficients,
typically set to be Gaussians, with the likelihood induced by probit or logit
regressions for the responses. In this updating, the apparent absence of a
tractable posterior has motivated a variety of computational methods, including
Markov Chain Monte Carlo routines and algorithms which approximate the
posterior. Despite being routinely implemented, Markov Chain Monte Carlo
strategies face mixing or time-inefficiency issues in large p and small n
studies, whereas approximate routines fail to capture the skewness typically
observed in the posterior. This article proves that the posterior distribution
for the probit coefficients has a unified skew-normal kernel, under Gaussian
priors. Such a novel result allows efficient Bayesian inference for a wide
class of applications, especially in large p and small-to-moderate n studies
where state-of-the-art computational methods face notable issues. These
advances are outlined in a genetic study, and further motivate the development
of a wider class of conjugate priors for probit models along with methods to
obtain independent and identically distributed samples from the unified
skew-normal posterior
Biological applications of the theory of birth-and-death processes
In this review, we discuss the applications of the theory of birth-and-death
processes to problems in biology, primarily, those of evolutionary genomics.
The mathematical principles of the theory of these processes are briefly
described. Birth-and-death processes, with some straightforward additions such
as innovation, are a simple, natural formal framework for modeling a vast
variety of biological processes such as population dynamics, speciation, genome
evolution, including growth of paralogous gene families and horizontal gene
transfer, and somatic evolution of cancers. We further describe how empirical
data, e.g., distributions of paralogous gene family size, can be used to choose
the model that best reflects the actual course of evolution among different
versions of birth-death-and-innovation models. It is concluded that
birth-and-death processes, thanks to their mathematical transparency,
flexibility and relevance to fundamental biological process, are going to be an
indispensable mathematical tool for the burgeoning field of systems biology.Comment: 29 pages, 4 figures; submitted to "Briefings in Bioinformatics
Dynamics of Barred Galaxies
Some 30% of disc galaxies have a pronounced central bar feature in the disc
plane and many more have weaker features of a similar kind. Kinematic data
indicate that the bar constitutes a major non-axisymmetric component of the
mass distribution and that the bar pattern tumbles rapidly about the axis
normal to the disc plane. The observed motions are consistent with material
within the bar streaming along highly elongated orbits aligned with the
rotating major axis. A barred galaxy may also contain a spheroidal bulge at its
centre, spirals in the outer disc and, less commonly, other features such as a
ring or lens. Mild asymmetries in both the light and kinematics are quite
common. We review the main problems presented by these complicated dynamical
systems and summarize the effort so far made towards their solution,
emphasizing results which appear secure. (Truncated)Comment: This old review appeared in 1993. Plain tex with macro file. 82 pages
18 figures. A pdf version with figures at full resolution (3.24MB) is
available at http://www.physics.rutgers.edu/~sellwood/bar_review.pd
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