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21
A Non-Markovian Coupling for Randomly Sampling Colorings
, 2005
"... 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 + ɛ)∆, for all ɛ& ..."
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Cited by 33 (6 self)
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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 + ɛ)∆, for all ɛ> 0, assuming g ≥ 11 and ∆ = Ω(log n). The best previously known bounds were k> 11∆/6 for general graphs, and k> 1.489 ∆ for graphs satisfying girth and maximum degree requirements. Our proof relies on the construction and analysis of a non-Markovian coupling. This appears to be the first application of a non-Markovian coupling to substantially improve upon known results.
Correlation decay and deterministic FPTAS for counting list-colorings of a graph
- In Proc. 18th ACM-SIAM Symp. Discret. Algorithms (2007), SIAM
"... Abstract We propose a deterministic algorithm for approximately counting the number of list colorings of a graph. Under the assumption that the graph is triangle free, the size of every list is at least α∆, where α is an arbitrary constant bigger than α * * = 2.8432 . . ., the solution of αe − 1 α ..."
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Cited by 30 (9 self)
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Abstract We propose a deterministic algorithm for approximately counting the number of list colorings of a graph. Under the assumption that the graph is triangle free, the size of every list is at least α∆, where α is an arbitrary constant bigger than α * * = 2.8432 . . ., the solution of αe − 1 α = 2, and ∆ is the maximum degree of the graph, we obtain the following results. For the case when the size of the each list is a large constant, we show the existence of a deterministic FPTAS for computing the total number of list colorings. The same deterministic algorithm has complexity 2 O(log 2 n) , without any assumptions on the sizes of the lists, where n is the size of the instance. Our results are not based on the most powerful existing counting technique -rapidly mixing Markov chain method. Rather we build upon concepts from statistical physics, in particular, the decay of correlation phenomena and its implication for the uniqueness of Gibbs measures in infinite graphs. This approach was proposed in two recent papers [BG06] and [Wei05]. The principle insight of the present work is that the correlation decay property can be established with respect to certain computation tree, as opposed to the conventional correlation decay property which is typically established with respect to graph theoretic neighborhoods of a given node. This allows truncation of computation at a logarithmic depth in order to obtain polynomial accuracy in polynomial time. While the analysis conducted in this paper is limited to the problem of counting list colorings, the proposed algorithm can be extended to an arbitrary constraint satisfaction problem in a straightforward way.
Counting without sampling. New algorithms for enumeration problems using statistical physics
- IN PROCEEDINGS OF SODA
, 2006
"... We propose a new type of approximate counting algorithms for the problems of enumerating the number of independent sets and proper colorings in low degree graphs with large girth. Our algorithms are not based on a commonly used Markov chain technique, but rather are inspired by developments in stati ..."
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Cited by 28 (6 self)
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We propose a new type of approximate counting algorithms for the problems of enumerating the number of independent sets and proper colorings in low degree graphs with large girth. Our algorithms are not based on a commonly used Markov chain technique, but rather are inspired by developments in statistical physics in connection with correlation decay properties of Gibbs measures and its implications to uniqueness of Gibbs measures on infinite trees, reconstruction problems and local weak convergence methods. On a negative side, our algorithms provide ǫ-approximations only to the logarithms of the size of a feasible set (also known as free energy in statistical physics). But on the positive side, our approach provides deterministic as opposed to probabilistic guarantee on approximations. Moreover, for some regular graphs we obtain explicit values for the counting problem. For example, we show that every 4-regular n-node graph with large girth has approximately (1.494...) n independent sets, and in every r-regular graph with n nodes and large girth the number of q ≥ r + 1-proper colorings is approximately [q(1 − 1 r q) 2] n, for large n. In statistical physics terminology, we compute explicitly the limit of the log-partition function. We extend our results to random regular graphs. Our explicit results would be hard to derive via the Markov chain method.
Counting Without Sampling: Asymptotics of the Log-Partition Function for Certain Statistical Physics Models
, 2006
"... In this article we propose new methods for computing the asymptotic value for the logarithm of the partition function (free energy) for certain statistical physics models on certain type of finite graphs, as the size of the underlying graph goes to infinity. The two models considered are the hard- ..."
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Cited by 28 (6 self)
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In this article we propose new methods for computing the asymptotic value for the logarithm of the partition function (free energy) for certain statistical physics models on certain type of finite graphs, as the size of the underlying graph goes to infinity. The two models considered are the hard-core (independent set) model when the activity parameter λ is small, and also the Potts (q-coloring) model. We only consider the graphs with large girth. In particular, we prove that asymptotically the logarithm of the number of independent sets of any r-regular graph with large girth when rescaled is approximately constant if r ≤ 5. For example, we show that every 4-regular n-node graph with large girth has approximately (1.494 ···) n-many independent sets, for large n. Further, we prove that for every r-regular graph with r ≥ 2, with n nodes and large girth, the number of proper q ≥ r +1 colorings is approximately [q(1 − 1 q) r 2] n, for large n. We also show that these results hold for random regular graphs with high probability (w.h.p.) as well. As a byproduct of our method we obtain simple algorithms for the problem of computing approximately the logarithm of the number of independent sets and proper colorings, in low degree
Randomly Coloring Graphs of Girth at Least Five (Extended Abstract)
- STOC'03
, 2003
"... We improve rapid mixing results for the simple Glauber dynamics designed to generate a random k-coloring of a bounded-degree graph. ..."
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Cited by 18 (3 self)
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We improve rapid mixing results for the simple Glauber dynamics designed to generate a random k-coloring of a bounded-degree graph.
A personal list of unsolved problems concerning Potts models and lattice gases
- TO APPEAR IN MARKOV PROCESSES AND RELATED FIELDS
, 2000
"... I review recent results and unsolved problems concerning the hard-core lattice gas and the q-coloring model (antiferromagnetic Potts model at zero temperature). For each model, I consider its equilibrium properties (uniqueness/nonuniqueness of the infinite-volume Gibbs measure, complex zeros of the ..."
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Cited by 17 (1 self)
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I review recent results and unsolved problems concerning the hard-core lattice gas and the q-coloring model (antiferromagnetic Potts model at zero temperature). For each model, I consider its equilibrium properties (uniqueness/nonuniqueness of the infinite-volume Gibbs measure, complex zeros of the partition function) and the dynamics of local and nonlocal Monte Carlo algorithms (ergodicity, rapid mixing, mixing at complex fugacity). These problems touch on mathematical physics, probability, combinatorics and theoretical computer science.
A simple condition implying rapid mixing of single-site dynamics on spin systems.
- In 47th Annual IEEE Symposium on Foundations of Computer Science,
, 2006
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Reconstruction for colorings on trees
, 2008
"... 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 i ..."
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Cited by 11 (4 self)
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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 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 non-vanishing influence on the root in expectation, over random colorings of the leaves, reconstruction is said to hold. Non-reconstruction 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 ∆, non-reconstruction 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.
