Polynomial-time approximation algorithms for the Ising model
SIAM Journal on Computing
On Counting Independent Sets in Sparse 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
Reconstruction for Models on Random Graphs
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Computational Transition at the Uniqueness Threshold
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Improved inapproximability results for counting independent sets in the hard-core model
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Improved Mixing Condition on the Grid for Counting and Sampling Independent Sets
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Approximate counting via correlation decay in spin systems
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Guest column: complexity dichotomies of counting problems
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In a seminal paper [12], Weitz gave a deterministic fully polynomial approximation scheme for counting exponentially weighted independent sets (equivalently, approximating the partition function of the hard-core model from statistical physics) on graphs of degree at most d, up to the critical activity for the uniqueness of the Gibbs measure on the infinite d-regular tree. More recently Sly [10] showed that this is optimal in the sense that if there is an FPRAS for the hard-core partition function on graphs of maximum degree d for activities larger than the critical activity on the infinite d-regular tree then NP = RP. In this paper, we extend Weitz's approach to derive a deterministic fully polynomial approximation scheme for the partition function of the anti-ferromagnetic Ising model with arbitrary field on graphs of maximum degree d, up to the corresponding critical point on the d-regular tree. The main ingredient of our result is a proof that for two-state anti-ferromagnetic spin systems on the d-regular tree, weak spatial mixing implies strong spatial mixing. This in turn uses a message-decay argument which extends a similar approach proposed recently for the hard-core model by Restrepo et al [9] to the case of the anti-ferromagnetic Ising model with arbitrary field. By a standard correspondence, these results translate to arbitrary two-state anti-ferromagnetic spin systems with soft constraints.