Results 1  10
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11
Counting independent sets up to the tree threshold
 In STOC ’06: Proceedings of the thirtyeighth annual ACM symposium on Theory of computing
, 2006
"... Consider the problem of approximately counting weighted independent sets of a graph G with activity λ, i.e., where the weight of an independent set I is λ I . We present a novel analysis yielding a deterministic approximation scheme which runs in polynomial time for any graph of maximum degree Δ ..."
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Cited by 51 (2 self)
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Consider the problem of approximately counting weighted independent sets of a graph G with activity λ, i.e., where the weight of an independent set I is λ I . We present a novel analysis yielding a deterministic approximation scheme which runs in polynomial time for any graph of maximum degree Δ and λ<λc =(Δ − 1) Δ−1 /(Δ − 2) Δ.Thisimproves on the previously known general bound of λ ≤ 2
Glauber dynamics on trees: boundary conditions and mixing time
 Comm. Math. Phys
"... We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the socalled Bethe approximation. Specifically, we show that spectral gap and the logSobolev constant of the Glauber dynamics for the Ising model on an nvertex regular tree ..."
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Cited by 23 (7 self)
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We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the socalled Bethe approximation. Specifically, we show that spectral gap and the logSobolev constant of the Glauber dynamics for the Ising model on an nvertex regular tree with (+)boundary are bounded below by a constant independent of n at all temperatures and all external fields. This implies that the mixing time is O(log n) (in contrast to the free boundary case, where it is not bounded by any fixed polynomial at low temperatures). In addition, our methods yield simpler proofs and stronger results for the spectral gap and logSobolev constant in the regime where there are multiple phases but the mixing time is insensitive to the boundary condition. Our techniques also apply to a much wider class of models, including those with hardcore constraints like the antiferromagnetic Potts model at zero temperature (proper colorings) and the hard–core lattice gas (independent sets).
Adaptive Simulated Annealing: A Nearoptimal Connection between Sampling and Counting
"... We present a nearoptimal reduction from approximately counting the cardinality of a discrete set to approximately sampling elements of the set. An important application of our work is to approximating the partition function Z of a discrete system, such as the Ising model, matchings or colorings of ..."
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Cited by 10 (5 self)
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We present a nearoptimal reduction from approximately counting the cardinality of a discrete set to approximately sampling elements of the set. An important application of our work is to approximating the partition function Z of a discrete system, such as the Ising model, matchings or colorings of a graph. The standard approach to estimating the partition function Z(β ∗ ) at some desired inverse temperature β ∗ is to define a sequence, which we call a cooling schedule, β0 =0<β1 < ·· · <βℓ = β ∗ where Z(0) is trivial to compute and the ratios Z(βi+1)/Z(βi) are easy to estimate by sampling from the distribution corresponding to Z(βi). Previous approaches required a cooling schedule of length O ∗ (ln A) where A = Z(0), thereby ensuring that each ratio Z(βi+1)/Z(βi) is bounded. We present a cooling schedule of length ℓ = O ∗ ( √ ln A). For wellstudied problems such as estimating the partition function of the Ising model, or approximating the number of colorings or matchings of a graph, our cooling schedule is of length O ∗ ( √ n) and the total number of samples required is O ∗ (n). This implies an overall savings of a factor of roughly n in the running time of the approximate counting algorithm compared to the previous best approach. A similar improvement in the length of the cooling schedule was recently obtained by Lovász and Vempala in the context of estimating the volume of convex bodies. While our reduction is inspired by theirs, the discrete analogue of their result turns out to be significantly more difficult. Whereas a fixed schedule suffices in their setting, we prove that in the discrete setting we need an adaptive schedule, i. e., the schedule depends on Z. More precisely, we prove any nonadaptive cooling schedule has length at least O ∗ (ln A), and we present an algorithm to find an adaptive schedule of length O ∗ ( √ ln A) and a nearly matching lower bound.
Reconstruction for colorings on tree
, 2008
"... Consider kcolorings 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 infinitevolume Gibbs measure. It i ..."
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Cited by 5 (2 self)
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Consider kcolorings 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 infinitevolume Gibbs measure. It is straightforward to show the existence of colorings of the leaves which “freeze ” the entire tree when k ≤ ∆ + 1. For k ≥ ∆ + 2, Jonasson proved the root is “unbiased ” for any fixed coloring of the leaves and thus the Gibbs measure is unique. What happens for a typical coloring of the leaves? When the leaves have a nonvanishing influence on the root in expectation, over random colorings of the leaves, reconstruction is said to hold. Nonreconstruction is equivalent to extremality of the Gibbs measure. When k < ∆ / ln ∆, it is straightforward to show that reconstruction is possible (and hence the measure is not extremal). We prove that for C> 2 and k = C∆ / ln ∆, nonreconstruction holds, i.e., the Gibbs measure is extremal. We prove a strong form of extremality: with high probability over the colorings of the leaves the influence at the root decays exponentially fast with the depth of the tree. These are the first results coming close to the threshold for extremality for colorings. Extremality on trees and random graphs has received considerable attention recently since it may have connections to the efficiency of local algorithms.
Strong spatial mixing with fewer colours for lattice graphs
 Proc. 45th IEEE Symp. on Foundations of Computer Science
"... Abstract Recursivelyconstructed couplings have been used in the past for mixing on trees. We show how to extend this technique to nontreelike graphs such as lattices. Using this method, we obtain the following general result. Suppose that G is a trianglefree graph and that for some \Delta * 3, ..."
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Cited by 5 (2 self)
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Abstract Recursivelyconstructed couplings have been used in the past for mixing on trees. We show how to extend this technique to nontreelike graphs such as lattices. Using this method, we obtain the following general result. Suppose that G is a trianglefree graph and that for some \Delta * 3, the maximum degree of G is at most \Delta. We show that the spin system consisting of qcolourings of G has strong spatial mixing, provided q? ff\Delta \Gamma fl, where ff ss 1:76322 is the solution to ff ff = e, and fl =
The Mixing Time of Glauber Dynamics for Colouring Regular Trees ∗
"... We consider Metropolis Glauber dynamics for sampling proper qcolourings of the nvertex complete bary tree when 3 ≤ q ≤ b/2 ln(b). We give both upper and lower bounds on the mixing time. For fixed q and b, our upper bound is n O(b / log b) and our lower bound is n Ω(b/q log(b)) , where the constan ..."
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Cited by 2 (0 self)
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We consider Metropolis Glauber dynamics for sampling proper qcolourings of the nvertex complete bary tree when 3 ≤ q ≤ b/2 ln(b). We give both upper and lower bounds on the mixing time. For fixed q and b, our upper bound is n O(b / log b) and our lower bound is n Ω(b/q log(b)) , where the constants implicit in the O() and Ω() notation do not depend upon n, q or b. 1
Modeling
, 1995
"... To Steve Schneider, whose interdisciplinary expertise and social responsibility as a climate modeler demanded philosophical recognition; and to Bill Wimsatt, whose philosophical analysis of biological complexity demanded application to the physical sciences. Contents ..."
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Cited by 1 (0 self)
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To Steve Schneider, whose interdisciplinary expertise and social responsibility as a climate modeler demanded philosophical recognition; and to Bill Wimsatt, whose philosophical analysis of biological complexity demanded application to the physical sciences. Contents
Improved mixing bounds for the AntiFerromagnetic Potts 2 ∗ Model on Z
, 2006
"... We consider the antiferromagnetic Potts model on the the integer lattice Z 2. The model has two parameters, q, the number of spins, and λ = exp(−β), where β is “inverse temperature”. It is known that the model has strong spatial mixing if q> 7, or if q = 7 and λ = 0 or λ> 1/8, or if q = 6 and ..."
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We consider the antiferromagnetic Potts model on the the integer lattice Z 2. The model has two parameters, q, the number of spins, and λ = exp(−β), where β is “inverse temperature”. It is known that the model has strong spatial mixing if q> 7, or if q = 7 and λ = 0 or λ> 1/8, or if q = 6 and λ = 0 or λ> 1/4. The λ = 0 case corresponds to the model in which configurations are proper qcolourings of Z 2. We show that the system has strong spatial mixing for q ≥ 6 and any λ. This implies that Glauber dynamics is rapidly mixing (so there is a fullypolynomial randomised approximation scheme for the partition function) and also that there is a unique infinitevolume Gibbs state. We also show that strong spatial mixing occurs for a larger range of λ than was previously known for q = 3, 4 and 5. 1 Introduction and statement of results 1.1 The antiferromagnetic Potts model We consider the antiferromagnetic Potts model on the integer lattice Z2. The set of spins is Q = {1,..., q}. Configurations are assignments of spins to vertices, and Ω = QZ2 is the set
AND
"... Abstract. We present a nearoptimal reduction from approximately counting the cardinality of a discrete set to approximately sampling elements of the set. An important application of our work is to approximating the partition function Z of a discrete system, such as the Ising model, matchings or col ..."
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Abstract. We present a nearoptimal reduction from approximately counting the cardinality of a discrete set to approximately sampling elements of the set. An important application of our work is to approximating the partition function Z of a discrete system, such as the Ising model, matchings or colorings of a graph. The typical approach to estimating the partition function Z(β ∗ ) at some desired inverse temperature β ∗ is to define a sequence, which we call a cooling schedule, β0 = 0 <β1 < ·· · <βℓ = β ∗ where Z(0) is trivial to compute and the ratios Z(βi+1)/Z(βi) are easy to estimate by sampling from the distribution corresponding to Z(βi). Previous approaches required a cooling schedule of length O ∗ (ln A) where A = Z(0), thereby ensuring that each ratio Z(βi+1)/Z(βi) is bounded. We present a cooling schedule of length ℓ = O ∗ ( √ ln A). For wellstudied problems such as estimating the partition function of the Ising model, or approximating the number of colorings or matchings of a graph, our cooling schedule is of length O ∗ ( √ n), which implies an overall savings of O ∗ (n) in the running time of the approximate counting algorithm (since roughly ℓ samples are needed to estimate each ratio). A similar improvement in the length of the cooling schedule was recently obtained by Lovász and Vempala in the context of estimating the volume of convex bodies. While our reduction is inspired by theirs, the discrete analogue of their result turns out to be significantly more difficult. Whereas a fixed schedule suffices in their setting, we prove that in the discrete setting we need an adaptive schedule, that is, the schedule depends on Z. More precisely, we prove any nonadaptive cooling schedule has length at least O ∗ (ln A), and we present an algorithm to find an adaptive schedule of length O ∗ ( √ ln A).
CONCENTRATION OF MEASURE AND MIXING FOR MARKOV CHAINS
, 809
"... Abstract. We consider Markovian models on graphs with local dynamics. We show that, under suitable conditions, such Markov chains exhibit both rapid convergence to equilibrium and strong concentration of measure in the stationary distribution. We illustrate our results with applications to some know ..."
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Abstract. We consider Markovian models on graphs with local dynamics. We show that, under suitable conditions, such Markov chains exhibit both rapid convergence to equilibrium and strong concentration of measure in the stationary distribution. We illustrate our results with applications to some known chains from computer science and statistical mechanics. 1.