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2012 IEEE 53rd Annual Symposium on Foundations of Computer Science (2012)
New Brunswick, NJ, USA USA
Oct. 20, 2012 to Oct. 23, 2012
ISSN: 0272-5428
ISBN: 978-1-4673-4383-1
pp: 361-369
The class of two-spin systems contains several important models, including random independent sets and the Ising model of statistical physics. We show that for both the hard-core (independent set) model and the anti-ferromagnetic Ising model with arbitrary external field, it is NP-hard to approximate the partition function or approximately sample from the model on regular graphs when the model has non-uniqueness on the corresponding regular tree. Together with results of Jerrum -- Sinclair, Weitz, and Sinclair -- Srivastava -- Thurley giving FPRAS's for all other two-spin systems except at the uniqueness threshold, this gives an almost complete classification of the computational complexity of two-spin systems on bounded-degree graphs. Our proof establishes that the normalized log-partition function of any two-spin system on bipartite locally tree-like graphs converges to a limiting ``free energy density'' which coincides with the (non-rigorous) Be the prediction of statistical physics. We use this result to characterize the local structure of two-spin systems on locally tree-like bipartite expander graphs, which then become the basic gadgets in a randomized reduction to approximate MAX-CUT. Our approach is novel in that it makes no use of the second moment method employed in previous works on these questions.
Bethe free energy, spin system, hard-core model, independent set, Ising model

A. Sly and N. Sun, "The Computational Hardness of Counting in Two-Spin Models on d-Regular Graphs," 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science(FOCS), New Brunswick, NJ, USA USA, 2012, pp. 361-369.
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