Inapproximability after Uniqueness Phase Transition in Two-Spin Systems

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


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 ZA(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 ZA(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 ZA(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 ZA(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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jin-Yi Cai
    • 1
  • Xi Chen
    • 2
  • Heng Guo
    • 1
  • Pinyan Lu
    • 3
  1. 1.University of WisconsinMadisonUSA
  2. 2.Columbia UniversityUSA
  3. 3.Microsoft Research AsiaUSA

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