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 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.


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Lovász, L.: Operations with structures. Acta Mathematica Hungarica 18, 321–328 (1967)zbMATHCrossRefGoogle Scholar
  2. 2.
    Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford University Press (2004)Google Scholar
  3. 3.
    Dyer, M., Greenhill, C.: The complexity of counting graph homomorphisms. In: Proceedings of the 9th International Conference on Random Structures and Algorithms, pp. 260–289 (2000)Google Scholar
  4. 4.
    Bulatov, A., Grohe, M.: The complexity of partition functions. Theoretical Computer Science 348(2-3), 148–186 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Goldberg, L., Grohe, M., Jerrum, M., Thurley, M.: A complexity dichotomy for partition functions with mixed signs. SIAM Journal on Computing 39(7), 3336–3402 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cai, J.Y., Chen, X., Lu, P.: Graph homomorphisms with complex values: A dichotomy theorem. In: Proceedings of the 37th Colloquium on Automata, Languages and Programming (2010); To appear in SIAM Journal on ComputingGoogle Scholar
  7. 7.
    Dyer, M., Goldberg, L., Paterson, M.: On counting homomorphisms to directed acyclic graphs. Journal of the ACM 54(6) (2007)Google Scholar
  8. 8.
    Cai, J.Y., Chen, X.: A decidable dichotomy theorem on directed graph homomorphisms with non-negative weights. In: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, pp. 437–446 (2010)Google Scholar
  9. 9.
    Bulatov, A.: The complexity of the counting constraint satisfaction problem. In: Proceedings of the 35th International Colloquium on Automata, Languages and Programming, pp. 646–661 (2008)Google Scholar
  10. 10.
    Dyer, M., Richerby, D.: On the complexity of #CSP. In: Proceedings of the 42nd ACM Symposium on Theory of Computing, pp. 725–734 (2010)Google Scholar
  11. 11.
    Cai, J.Y., Chen, X., Lu, P.: Non-negatively weighted #CSP: An effective complexity dichotomy. In: Proceedings of the 26th Annual IEEE Conference on Computational Complexity, pp. 45–54 (2011)Google Scholar
  12. 12.
    Cai, J.Y., Chen, X.: Complexity of counting CSP with complex weights. In: Proceedings of the 44th ACM Symposium on Theory of Computing (2012)Google Scholar
  13. 13.
    Jerrum, M., Sinclair, A.: Polynomial-time approximation algorithms for the ising model. SIAM Journal on Computing 22(5), 1087–1116 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Goldberg, L., Jerrum, M., Paterson, M.: The computational complexity of two-state spin systems. Random Structures and Algorithms 23(2), 133–154 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Dyer, M., Frieze, A., Jerrum, M.: On counting independent sets in sparse graphs. SIAM Journal on Computing 31, 1527–1541 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Weitz, D.: Counting independent sets up to the tree threshold. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp. 140–149 (2006)Google Scholar
  17. 17.
    Sinclair, A., Srivastava, P., Thurley, M.: Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (2012)Google Scholar
  18. 18.
    Li, L., Lu, P., Yin, Y.: Approximate counting via correlation decay in spin systems. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (2012)Google Scholar
  19. 19.
    Li, L., Lu, P., Yin, Y.: Correlation decay up to uniqueness in spin systems. arXiv:1111.7064 (2011)Google Scholar
  20. 20.
    Bandyopadhyay, A., Gamarnik, D.: Counting without sampling: Asymptotics of the log-partition function for certain statistical physics models. Random Structures and Algorithms 33, 452–479 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Sly, A.: Computational transition at the uniqueness threshold. In: Proceedings of the IEEE 51st Annual Symposium on Foundations of Computer Science (2010)Google Scholar
  22. 22.
    Galanis, A., Ge, Q., Štefankovič, D., Vigoda, E., Yang, L.: Improved inapproximability results for counting independent sets in the hard-core model. In: Proceedings of the 15th International Workshop on Randomization and Computation, pp. 567–578 (2011)Google Scholar
  23. 23.
    Cai, J.Y., Chen, X., Guo, H., Lu, P.: Inapproximability after uniqueness phase transition in two-spin systems. University of Wisconsin, Madison CS Technical Report (2011),
  24. 24.
    Mossel, E., Weitz, D., Wormald, N.: On the hardness of sampling independent sets beyond the tree threshold. Probability Theory and Related Fields 143, 401–439 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Håstad, J.: Some optimal inapproximability results. Journal of the ACM 48, 798–859 (2001)MathSciNetzbMATHCrossRefGoogle Scholar

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

Personalised recommendations