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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)

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)

- 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)

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)

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 G… (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)

We analyze Markov chains for generating a random k-coloring of a random graph G n,d/n. When the average degree d is constant, a random graph has maximum degree log n/ log log n, with high probability. We efficiently generate a random k-coloring when k = Ω(log log n/ log log log n), i.e., with many fewer colors than the maximum degree. Previous results hold… (More)

— Given n elements with non-negative integer weights w1,. .. , wn and an integer capacity C, we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most C. We give the first de-terministic, fully polynomial-time approximation scheme (FPTAS) for estimating the number of solutions to… (More)

Consider k-colorings of the complete tree of depth ℓ and branching factor ∆. If we fix the coloring of the leaves, for what range of k is the root uniformly distributed over all k colors (in the limit ℓ → ∞)? This corresponds to the threshold for uniqueness of the infinite-volume Gibbs measure. It is straightforward to show the existence of colorings of the… (More)