The computational hardness of counting in two-spin models on d-regular graphs

TL;DR

This paper proves NP-hardness of approximating partition functions and sampling in two-spin models on d-regular graphs at non-uniqueness, completing the complexity classification for these models and connecting statistical physics predictions to computational hardness.

Contribution

It establishes the NP-hardness of approximation and sampling for two-spin systems at non-uniqueness, and links the Bethe free energy to the local structure of these models.

Findings

NP-hardness results for hard-core and anti-ferromagnetic Ising models at non-uniqueness

Convergence of normalized log-partition function to Bethe free energy

Characterization of local structure on bipartite expander graphs

Abstract

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 d-regular graphs when the model has non-uniqueness on the d-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) Bethe 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.