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Consider the problem of approximately counting weighted independent sets of a graph G with activity &#955;, i.e., where the weight of an independent set I is &#955;<sup>|I|</sup>. We present a novel analysis yielding a deterministic approximation scheme which runs in polynomial time for <i>any</i> graph of maximum degree &#916; and &#955;&lt;(More)
We present a random walk as an eÆcient and accurate approach to approximating certain aggregate queries about web pages. Our method uses a novel random walk to produce an almost uniformly distributed sample of web pages. The walk traverses a dynamically built regular undirected graph. Queries we have estimated using this method include the coverage of(More)
We consider local Markov chain Monte-Carlo algorithms for sampling from the weighted distribution of independent sets with activity λ, where the weight of an independent set I is λ. A recent result has established that Gibbs sampling is rapidly mixing in sampling the distribution for graphs of maximum degree d and λ < λc(d), where λc(d) is the critical(More)
We generalize previously known conditions for uniqueness of the Gibbs measure in statistical physics models by presenting conditions of any finite size for models on any underlying graph. We give two dual conditions, one requiring that the total influence on a site is small, and the other that the total influence of a site is small. Our proofs are(More)
We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the so-called Bethe approximation. Specifically, we show that spectral gap and the log-Sobolev constant of the Glauber dynamics for the Ising model on an n-vertex regular tree with (+)-boundary are bounded below by a constant(More)
We study the mixing time of the Glauber dynamics for general spin systems on bounded-degree trees, including the Ising model, the hard-core model (independent sets) and the antiferromagnetic Potts model at zero temperature (colorings). We generalize a framework, developed in our recent paper [18] in the context of the Ising model, for establishing mixing(More)
The paper considers spin systems on the d-dimensional integer lattice Z with nearestneighbor 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 show(More)
Foreword These are scribed notes from a graduate courses on Computational Complexity offered at the University of California at Berkeley in the Fall of 2002, based on notes scribed by students in Spring 2001 and on additional notes scribed in Fall 2002. I added notes and references in May 2004. The first 15 lectures cover fundamental material. The remaining(More)
We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics for the Ising model. Specifically, we show that the mixing time on an -vertex regular tree with -boundary remains at all temperatures (in contrast to the free boundary case, where the mixing time is not bounded by any fixed polynomial at(More)