The Markov chain Monte Carlo method: an approach to approximate counting and integration
Approximation algorithms for NP-hard problems
The Number of Independent Sets in a Grid Graph
SIAM Journal on Discrete Mathematics
The ζ (2) limit in the random assignment problem
Random Structures & Algorithms
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
Combinatorics, Probability and Computing
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One of the key computational problems in combinatorics/statistical physics is the problem of computing limits of the log-partition functions for various statistical mechanics models on lattices. In combinatorics this limit corresponds to the exponent of various arrangements on lattices, for example the exponents of the number of independent sets, proper colorings or matchings on a lattice. In statistical physics this limit is called free energy. We propose a new method, sequential cavity, which beats the best known existing methods, such as transfer matrix method, in obtaining sharper bounds on the limits of the log-partition function for two models: independent sets (hard-core) and matchings (monomer-dimer). Our method is based on a surprisingly simple representation of the log-partition function limit in terms of a certain marginal probability of a suitably modified lattice, and using recent deterministic approximation counting algorithms for these two models. Our method also has a provably better theoretical performance compared with the transfer matrix method.