16,598 research outputs found
k-Center Clustering Under Perturbation Resilience
The k-center problem is a canonical and long-studied facility location and clustering problem with many applications in both its symmetric and asymmetric forms. Both versions of the problem have tight approximation factors on worst case instances: a 2-approximation for symmetric kcenter and an O(log*(k))-approximation for the asymmetric version. Therefore to improve on these ratios, one must go beyond the worst case.
In this work, we take this approach and provide strong positive results both for the asymmetric and symmetric k-center problems under a very natural input stability (promise) condition called alpha-perturbation resilience [Bilu Linial, 2012], which states that the optimal solution does not change under any alpha-factor perturbation to the input distances. We show that by assuming 2-perturbation resilience, the exact solution for the asymmetric k-center problem can be found in polynomial time. To our knowledge, this is the first problem that is hard to approximate to any constant factor in the worst case, yet can be optimally solved in polynomial time under perturbation resilience for a constant value of alpha. Furthermore, we prove our result is tight by showing symmetric k-center under (2-epsilon)-perturbation resilience is hard unless NP=RP.
This is the first tight result for any problem under perturbation resilience, i.e., this is the first time the exact value of alpha for which the problem switches from being NP-hard to efficiently computable has been found.
Our results illustrate a surprising relationship between symmetric and asymmetric k-center instances under perturbation resilience. Unlike approximation ratio, for which symmetric k-center is easily solved to a factor of 2 but asymmetric k-center cannot be approximated to any constant factor, both symmetric and asymmetric k-center can be solved optimally under resilience
to 2-perturbations
Levy--Brownian motion on finite intervals: Mean first passage time analysis
We present the analysis of the first passage time problem on a finite
interval for the generalized Wiener process that is driven by L\'evy stable
noises. The complexity of the first passage time statistics (mean first passage
time, cumulative first passage time distribution) is elucidated together with a
discussion of the proper setup of corresponding boundary conditions that
correctly yield the statistics of first passages for these non-Gaussian noises.
The validity of the method is tested numerically and compared against
analytical formulae when the stability index approaches 2, recovering
in this limit the standard results for the Fokker-Planck dynamics driven by
Gaussian white noise.Comment: 9 pages, 13 figure
Instability of the symmetric Couette-flow in a granular gas: hydrodynamic field profiles and transport
We investigate the inelastic hard disk gas sheared by two parallel bumpy
walls (Couette-flow). In our molecular dynamic simulations we found a
sensitivity to the asymmetries of the initial condition of the particle places
and velocities and an asymmetric stationary state, where the deviation from
(anti)symmetric hydrodynamic fields is stronger as the normal restitution
coefficient decreases. For the better understanding of this sensitivity we
carried out a linear stability analysis of the former kinetic theoretical
solution [Jenkins and Richman: J. Fluid. Mech. {\bf 171} (1986)] and found it
to be unstable. The effect of this asymmetry on the self-diffusion coefficient
is also discussed.Comment: 9 pages RevTeX, 14 postscript figures, sent to Phys. Rev.
Active Brownian Particles. From Individual to Collective Stochastic Dynamics
We review theoretical models of individual motility as well as collective
dynamics and pattern formation of active particles. We focus on simple models
of active dynamics with a particular emphasis on nonlinear and stochastic
dynamics of such self-propelled entities in the framework of statistical
mechanics. Examples of such active units in complex physico-chemical and
biological systems are chemically powered nano-rods, localized patterns in
reaction-diffusion system, motile cells or macroscopic animals. Based on the
description of individual motion of point-like active particles by stochastic
differential equations, we discuss different velocity-dependent friction
functions, the impact of various types of fluctuations and calculate
characteristic observables such as stationary velocity distributions or
diffusion coefficients. Finally, we consider not only the free and confined
individual active dynamics but also different types of interaction between
active particles. The resulting collective dynamical behavior of large
assemblies and aggregates of active units is discussed and an overview over
some recent results on spatiotemporal pattern formation in such systems is
given.Comment: 161 pages, Review, Eur Phys J Special-Topics, accepte
Possible Implications of Asymmetric Fermionic Dark Matter for Neutron Stars
We consider the implications of fermionic asymmetric dark matter for a "mixed
neutron star" composed of ordinary baryons and dark fermions. We find examples,
where for a certain range of dark fermion mass -- when it is less than that of
ordinary baryons -- such systems can reach higher masses than the maximal
values allowed for ordinary ("pure") neutron stars. This is shown both within a
simplified, heuristic Newtonian analytic framework with non-interacting
particles and via a general relativistic numerical calculation, under certain
assumptions for the dark matter equation of state. Our work applies to various
dark fermion models such as mirror matter models and to other models where the
dark fermions have self interactions.Comment: 20 pages, 6 figure
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