The Markov chain Monte Carlo method: an approach to approximate counting and integration
Approximation algorithms for NP-hard problems
Strong Spatial Mixing with Fewer Colors for Lattice Graphs
SIAM Journal on Computing
Counting independent sets up to the tree threshold
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Simple deterministic approximation algorithms for counting matchings
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Correlation decay and deterministic FPTAS for counting list-colorings of a graph
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Random Structures & Algorithms
A deterministic approximation algorithm for computing the permanent of a 0,1 matrix
Journal of Computer and System Sciences
A very simple algorithm for estimating the number of k‐colorings of a low‐degree graph
Random Structures & Algorithms
Loopy belief propagation and Gibbs measures
UAI'02 Proceedings of the Eighteenth conference on Uncertainty in artificial intelligence
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We construct a deterministic algorithm for approximately counting the number of colorings of a graph. Under the assumption that the graph is triangle-free and the number of colors is at least @a@D, where @a is an arbitrary constant bigger than @a^@?^@?=2.8432...@?, and @D is the maximum degree of the graph, we obtain the following results. For the case when the number of colors is a large constant, we prove the existence of a deterministic FPTAS for computing the total number of colorings. The same deterministic algorithm has complexity 2^O^(^l^o^g^^^2^n^), without any assumptions on the number of colors, where n is the instance size. We further extend our method to the general problem of computing the partition function of a discrete Markov random field (MRF) model. Under certain assumptions relating the cardinality of the alphabet, the degree of the graph and the interacting potentials, we construct a deterministic FPTAS for computing the partition function of a MRF. In contrast to our results, the known counting technique - rapidly mixing Markov chain method - is based on randomization. Thus our result is the first non-trivial deterministic FPTAS for the problem of counting the number of colorings. Our approach builds on a certain statistical physics concept, specifically, 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 by Bandyopadhyay and Gamarnik (2008) [1] and Weitz (2006) [25]. The main distinction of this work is that we establish the correlation decay property on a computation tree arising from a certain recursive procedure, rather than reducing the problem to the one on a self-avoiding tree of a graph, as is done in Weitz (2006) [25]. This lets us deal with problems with more than two colors, which the tree-based approach of Weitz (2006) [25] is not capable of solving.