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We present a polynomial-time randomized algorithm for estimating the permanent of an arbitrary <i>n</i> × <i>n</i> matrix with nonnegative entries. This algorithm---technically a "fully-polynomial randomized approximation scheme"---computes an approximation that is, with high probability, within arbitrarily small specified relative error of the true… (More)

We consider the problem of sampling independent sets of a graph with maximum degree. The weight of each independent set is expressed in terms of a xed positive parameter 2 ?2 , where the weight of an independent set is jj. The Glauber dynamics is a simple Markov chain Monte Carlo method for sampling from this distribution. We show fast convergence of this… (More)

We presenta fully-polynomial scheme to approximate the number of independentsets in graphs with maximum degree four. In general, for graphswith maximum degree A ~ 4, the scheme approximates a weighted sum of independent sets. The weight of each independent set is expressed in terms of a positive parameter A < ~, where the weight of independent set S is… (More)

The paper considers spin systems on the d-dimensional integer lattice Z d with nearest-neighbor interactions. A sharp equivalence is proved between decay with distance of spin correlations (a spatial property of the equilibrium state) and rapid mixing of the Glauber dynamics (a temporal property of a Markov chain Monte Carlo algorithm). Specifically, we… (More)

This note considers the problem of sampling from the set of weighted independent sets of a graph with maximum degree. For a positive fugacity , the weight of an independent set is jj. Luby and Vigoda proved that the Glauber dynamics , which only changes the connguration at a randomly chosen vertex in each step, has mixing time O(n log n) when < 2 ?2 for… (More)

We present an improved coupling technique for analyzing the mixing time of Markov chains. Using our technique, we simplify and extend previous results for sampling colorings and independent sets. Our approach uses properties of the stationary distribution to avoid worst-case configurations which arise in the traditional approach.As an application, we show… (More)

- Eric Vigoda
- FOCS
- 1999

- Christian Borgs, Jennifer T. Chayes, Alan M. Frieze, Jeong Han Kim, Prasad Tetali, Eric Vigoda +1 other
- FOCS
- 1999

We study two widely used algorithms, Glauber dynamics and the Swendsen-Wang algorithm, on rectangular subsets of the hypercubic lattice Z d. We prove that under certain circumstances, the mixing time in a box of side length L with periodic boundary conditions can be exponential in L d−1. In other words, under these circumstances, the mixing in these widely… (More)

We study a simple Markov chain, known as the Glauber dynamics , for randomly sampling (proper) k-colorings of an input graph G on n vertices with maximum degree ∆ and girth g. We prove the Glauber dynamics is close to the uniform distribution after O(n log n) steps whenever k > (1 +)∆, for all > 0, assuming g ≥ 11 and ∆ = Ω(log n). The best previously known… (More)

The hard-core model has received much attention in the past couple of decades as a lattice gas model with hard constraints in statistical physics, a multicast model of calls in communication networks, and as a weighted independent set problem in combinatorics, probability and theoretical computer science. In this model, each independent set $I$ in a graph… (More)