# Inapproximability after Uniqueness Phase Transition in Two-Spin Systems

• Jin-Yi Cai
• Xi Chen
• Heng Guo
• Pinyan Lu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7402)

## 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.

## Keywords

Partition Function Bipartite Graph Regular Graph Gibbs Measure Input Graph
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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• Jin-Yi Cai
• 1
• Xi Chen
• 2
• Heng Guo
• 1
• Pinyan Lu
• 3