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22
The Computational Complexity Of TwoState Spin Systems
"... The subject of this article is spinsystems as studied in statistical physics. We focus on the case of two spins. This case encompasses models of physical interest, such as the classical Ising model (ferromagnetic or antiferromagnetic, with or without an applied magnetic field) and the hardcore ..."
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Cited by 22 (3 self)
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The subject of this article is spinsystems as studied in statistical physics. We focus on the case of two spins. This case encompasses models of physical interest, such as the classical Ising model (ferromagnetic or antiferromagnetic, with or without an applied magnetic field) and the hardcore gas model. There are three degrees of freedom, corresponding to our parameters , "/ and/. We wish to study the complexity of (approximately) computing the partition function in terms of these parameters. We pay special attention to the symmetric case/ = 1 for which our results are depicted in Figure 1. Exact computation of the partition function Z is NPhard except in the trivial case fi"/ = 1, so we concentrate on the issue of whether Z can be computed within small relative error in polynomial time. We show that there is a fully polynomial randomised approximation scheme (FPRAS) for the partition function in the "ferromagnetic" region "/ _> 1, but (unless RP = NP) there is no FPRAS in the "antiferromagnetic" region corresponding to the square defined by 0 < < 1 and 0 < ' < 1. Neither of these "natural" regions  neither the hyperbola nor the square  marks the boundary between tractable and intractable. In one direction, we provide an FPRAS for the partition function within a region which extends well away from the hyperbola. In the other direction, we exhibit two tiny, symmetric, intractable regions extending beyond the antiferromagnetic region. We also extend our results to the asymmetric case/ 1.
An extension of path coupling and its application to the Glauber dynamics for graph colourings
, 2000
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Approximate counting via correlation decay in spin systems
 In Proceedings of the 23rd Annual ACMSIAM Symposium on Discrete Algorithms
, 2012
"... We give the first deterministic fully polynomialtime approximation scheme (FPTAS) for computing the partition function of a twostate spin system on an arbitrary graph, when the parameters of the system satisfy the uniqueness condition on infinite regular trees. This condition is of physical signif ..."
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We give the first deterministic fully polynomialtime approximation scheme (FPTAS) for computing the partition function of a twostate spin system on an arbitrary graph, when the parameters of the system satisfy the uniqueness condition on infinite regular trees. This condition is of physical significance and is believed to be the right boundary between approximable and inapproximable. The FPTAS is based on the correlation decay technique introduced by Bandyopadhyay and Gamarnik [1] and Weitz [61]. The classic correlation decay is defined with respect to graph distance. Although this definition has natural physical meanings, it does not directly support an FPTAS for systems on arbitrary graphs, because for graphs with unbounded degrees, the local computation that provides a desirable precision by correlation decay may take superpolynomial time. We introduce a notion of computationally efficient correlation decay, in which the correlation decay is measured in a refined metric instead of graph distance. We use a potential method to analyze the amortized behavior of this correlation decay and establish a correlation decay that guarantees an inversepolynomial precision by polynomialtime local computation. This gives us an FPTAS for spin systems on arbitrary graphs. This new notion of correlation decay properly reflects the algorithmic aspect of the spin systems, and may be used for designing FPTAS for other counting problems. 1
Very rapid mixing of the Glauber dynamics for proper colourings on boundeddegree graphs
, 2000
"... Recent results have shown that the Glauber dynamics for graph colourings has optimal mixing time when (i) the graph is trianglefree and Dregular and the number of colours k is a small constant fraction smaller than 2D, or (ii) the graph has maximum degree D and k = 2D. We extend both these results ..."
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Cited by 11 (2 self)
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Recent results have shown that the Glauber dynamics for graph colourings has optimal mixing time when (i) the graph is trianglefree and Dregular and the number of colours k is a small constant fraction smaller than 2D, or (ii) the graph has maximum degree D and k = 2D. We extend both these results to prove that the Glauber dynamics has optimal mixing time when the graph has maximum degree D and the number of colours is a small constant fraction smaller than 2D.
Quantitative common carotid artery blood flow: prediction of internal carotid artery stenosis
 Magn. Reson. Imaging 4 37–42 Brands P J, Hoeks A P, Hofstra L and Reneman R S
, 1986
"... We consider Glauber dynamics on finite spin systems. The mixing time of Glauber dynamics can be bounded in terms of the influences of sites on each other. We consider three parameters bounding these influences — α, the total influence on a site, as studied by Dobrushin; α ′ , the total influence of ..."
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We consider Glauber dynamics on finite spin systems. The mixing time of Glauber dynamics can be bounded in terms of the influences of sites on each other. We consider three parameters bounding these influences — α, the total influence on a site, as studied by Dobrushin; α ′ , the total influence of a site, as studied by Dobrushin and Shlosman; and α ′ ′ , the total influence of a site in any given context, which is related to the pathcoupling method of Bubley and Dyer. It is known that if any of these parameters is less than 1 then randomupdate Glauber dynamics (in which a randomlychosen site is updated at each step) is rapidly mixing. It is also known that the Dobrushin condition α < 1 implies that systematicscan Glauber dynamics (in which sites are updated in a deterministic order) is rapidly mixing. This paper studies two related issues, primarily in the context of systematic scan: (1) the relationship between the parameters α, α ′ and α ′ ′ , and (2) the relationship between proofs of rapid mixing using Dobrushin uniqueness (which typically use analysis techniques) and proofs of rapid mixing using path coupling. We use matrixbalancing to show that the DobrushinShlosman condition α ′ < 1 implies rapid mixing of systematic scan. An interesting question is whether the rapid mixing results for scan can be extended to the α = 1 or α ′ = 1 case. We give positive results for the rapid mixing of systematic scan for certain α = 1 cases. As an application, we show rapid mixing of systematic scan (for any scan order) for heatbath Glauber dynamics for proper qcolourings of a degree ∆ graph G when q ≥ 2∆. 1
Counting and sampling Hcolourings
, 2003
"... For counting problems in #P which are “essentially selfreducible”, it is known that sampling and approximate counting are equivalent. However, many problems of interest do not have such a structure and there is already some evidence that this equivalence does not hold for the whole of #P. An intrig ..."
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Cited by 7 (2 self)
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For counting problems in #P which are “essentially selfreducible”, it is known that sampling and approximate counting are equivalent. However, many problems of interest do not have such a structure and there is already some evidence that this equivalence does not hold for the whole of #P. An intriguing example is the class of Hcolouring problems, which have recently been the subject of much study, and their natural generalisation to vertex and edgeweighted versions. Particular cases of the countingtosampling reduction have been observed, but it has been an open question as to how far these reductions might extend to any H and a general graph G. Here we give the first completely general countingtosampling reduction. For every fixed H, we show that the problem of approximately determining the partition function of weighted Hcolourings can be reduced to the problem of sampling these colourings from an approximately correct distribution. In particular, any rapidlymixing Markov chain for sampling Hcolourings can be turned into an FPRAS for counting Hcolourings.
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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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 =
On Markov chains for randomly Hcolouring a graph
, 2000
"... Let H = (W; F ) be a graph without multiple edges, but with the possibility of having loops. Let G = (V; E) be a simple graph. A homomorphism c is a map c : V !W with the property that (v; w) 2 E implies (c(v); c(w)) 2 F . We will often refer to c(v) as the colour of v and c as an Hcolouring of ..."
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Cited by 6 (3 self)
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Let H = (W; F ) be a graph without multiple edges, but with the possibility of having loops. Let G = (V; E) be a simple graph. A homomorphism c is a map c : V !W with the property that (v; w) 2 E implies (c(v); c(w)) 2 F . We will often refer to c(v) as the colour of v and c as an Hcolouring of G. We consider the problem of choosing a random Hcolouring of G by Markov Chain Monte Carlo. The probabilistic model we consider includes random proper colourings, random independent sets and the WidomRowlinson and Beach models of Statistical Physics. We prove negative results for uniform sampling and a positive result for weighted sampling when H is a tree. 1 Introduction We consider a class of graph labellings which are the natural generalisation of wellstudied problems such as proper kcolourings and independent sets. School of Mathematical Sciences, University of North London,London N7 8DB,UK y School of Computer Studies, University of Leeds, Leeds LS2 9TJ, UK z Department of...
Sampling grid colourings with fewer colours
 PROC. OF LATIN ’04
, 2004
"... We provide an optimally mixing Markov chain for 6colourings of the square grid. Furthermore, this implies that the uniform distribution on the set of such colourings has strong spatial mixing. 4 and 5 are now the only remaining values of k for which it is not known whether there exists a rapidly mi ..."
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We provide an optimally mixing Markov chain for 6colourings of the square grid. Furthermore, this implies that the uniform distribution on the set of such colourings has strong spatial mixing. 4 and 5 are now the only remaining values of k for which it is not known whether there exists a rapidly mixing Markov chain for kcolourings of the square grid.
The complexity of choosing an Hcolouring (nearly) uniformly at random
, 2003
"... Cooper, Dyer and Frieze studied the problem of sampling Hcolourings (nearly) uniformly at random. Special cases of this problem include sampling colourings and independent sets and sampling from statistical physics models such as the WidomRowlinson model, the Beach model, the Potts model and the h ..."
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Cited by 5 (1 self)
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Cooper, Dyer and Frieze studied the problem of sampling Hcolourings (nearly) uniformly at random. Special cases of this problem include sampling colourings and independent sets and sampling from statistical physics models such as the WidomRowlinson model, the Beach model, the Potts model and the hardcore lattice gas model. Cooper et al. considered the family of “cautious ” ergodic Markov chains with uniform stationary distribution and showed that, for every fixed connected “nontrivial” graph H, every such chain mixes slowly. In this paper, we give a complexity result for the problem. Namely, we show that for any fixed graph H with no trivial components, there is unlikely to be any Polynomial Almost Uniform Sampler (PAUS) for Hcolourings. We show that if there were a PAUS for the Hcolouring problem, there would also be a PAUS for sampling independent sets in bipartite graphs and, by the selfreducibility of the latter problem, there would be a FullyPolynomial Randomised Approximation