Inapproximability after Uniqueness Phase Transition in Two-Spin Systems

Abstract

A two-state spin system is specified by a matrix

${{\rm A}= \begin{bmatrix} \hspace{0.08cm} A_{0,0} & A_{0,1}\hspace{0.08cm} \\ \hspace{0.08cm}A_{1,0} & A_{1,1}\hspace{0.08cm} \end{bmatrix} = \begin{bmatrix} \hspace{0.08cm}\beta & 1\hspace{0.08cm} \\ \hspace{0.08cm}1 & \gamma\hspace{0.08cm} \end{bmatrix} } ~~~ (1)$

where β, γ ≥ 0. Given an input graph G = (V,E), the partition function Z _A (G) of a system is defined as

$Z_{\bf A}(G)= \sum\limits_{\sigma: V\rightarrow \{0, 1\}} \prod\limits_{(u,v) \in E} A_{\sigma(u), \sigma(v)}.~~ (2)$

We prove inapproximability results for the partition function Z _A (G) in the region specified by the non-uniqueness condition from phase transition for the Gibbs measure. More specifically, assuming $\text{{NP}} \not=\text{{RP}}$, for any fixed β,γ in the unit square, there is no randomized polynomial-time algorithm that approximates Z _A (G) for d-regular graphs G with relative error ε = 10^− 4, if d = Ω(Δ(β,γ)), where Δ(β,γ) > 1/(1 − βγ) is the uniqueness threshold. Up to a constant factor, this hardness result confirms the conjecture that the uniqueness phase transition coincides with the transition from computational tractability to intractability for Z _A (G). We also show a matching inapproximability result for a region of parameters β,γ outside the unit square, and all our results generalize to partition functions with an external field.