# Approximation Algorithms for Two-State Anti-Ferromagnetic Spin Systems on Bounded Degree Graphs

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## Abstract

We show that for the anti-ferromagnetic Ising model on the Bethe lattice, weak spatial mixing implies strong spatial mixing. As a by-product of our analysis, we obtain what is to the best of our knowledge the first rigorous proof of the uniqueness threshold for the anti-ferromagnetic Ising model (with non-zero external field) on the Bethe lattice. Following a method due to Weitz [15], we then use the equivalence between weak and strong spatial mixing to give a deterministic fully polynomial time approximation scheme for the partition function of the anti-ferromagnetic Ising model with arbitrary field on graphs of degree at most \(d\), throughout the uniqueness region of the Gibbs measure on the infinite \(d\)-regular tree. By a standard correspondence, our results translate to arbitrary two-state anti-ferromagnetic spin systems with soft constraints. Subsequent to a preliminary version of this paper, Sly and Sun [13] have shown that our results are optimal in the sense that, under standard complexity theoretic assumptions, there does not exist a fully polynomial time approximation scheme for the partition function of such spin systems on graphs of maximum degree \(d\) for parameters outside the uniqueness region. Taken together, the results of [13] and of this paper therefore indicate a tight relationship between complexity theory and phase transition phenomena in two-state anti-ferromagnetic spin systems.

## Keywords

Phase transitions Complexity theory Approximation algorithms Decay of correlations Two-spin systems on trees## Notes

### Acknowledgments

We thank Prasad Tetali for providing a manuscript of [11]. We also thank Colin McQuillan, Dror Weitz, Yitong Yin and two anonymous referees for several helpful comments. Alistair Sinclair was supported in part by United States National Science Foundation (NSF) Grant CCF-1016896. Piyush Srivastava was supported by the Berkeley Fellowship for Graduate Study and by NSF grant CCF-1016896, and performed part of this work while he was a research intern at Microsoft Research India. Marc Thurley was supported in part by a postdoctoral fellowship of the German Academic Exchange Service (DAAD) and by Marie Curie Intra-European Fellowship 271959, and performed part of this work while he was a postdoctoral scholar at the University of California, Berkeley, and at the Centre de Recerca Mathemàtica, Bellaterra.

## References

- 1.Arora, S., Barak, B.: Computational complexity: A modern approach. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
- 2.Dyer, M.E., Frieze, A.M., Jerrum, M.: On counting independent sets in sparse graphs. SIAM J. Comput.
**31**(5), 1527–1541 (2002)CrossRefzbMATHMathSciNetGoogle Scholar - 3.Galanis, A., Ge, Q., Stefankovic, D., Vigoda, E., Yang, L.: Improved inapproximability results for counting independent sets in the hard-core model. Proceedings of 14th International Workshop and 15th International Conference on Approximation, Randomization, and Combinatorial Optimization (APRROX-RANDOM), pp. 567–578. Springer-Verlag, Berlin (2011)Google Scholar
- 4.Georgii, H.O.: Gibbs measures and phase transitions. Walter de Gruyter Inc., De Gruyter Studies in Mathematics, New York (1988)CrossRefzbMATHGoogle Scholar
- 5.Gerschenfeld, A., Montanari, A.: Reconstruction for models on random graphs. Proceedings of 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 194–204. IEEE Computer Society, Washington, DC (2007)CrossRefGoogle Scholar
- 6.Goldberg, L.A., Jerrum, M., Paterson, M.: The computational complexity of two-state spin systems. Random Struct. Algorithms
**23**, 133–154 (2003)CrossRefzbMATHMathSciNetGoogle Scholar - 7.Jerrum, M., Sinclair, A.: Polynomial-time approximation algorithms for the Ising model. SIAM J. Comput.
**22**(5), 1087–1116 (1993)CrossRefzbMATHMathSciNetGoogle Scholar - 8.Li, L., Lu, P., Yin, Y.: Correlation decay up to uniqueness in spin systems. Proceedings of 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 67–84. SIAM, Philadelphia (2013)Google Scholar
- 9.Li, L., Lu, P., Yin, Y.: Approximate counting via correlation decay in spin systems. Proceedings of 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 922–940. SIAM, Philadelphia (2012)CrossRefGoogle Scholar
- 10.Mossel, E., Weitz, D., Wormald, N.: On the hardness of sampling independent sets beyond the tree threshold. Probab. Theory Relat. Fields
**143**(3–4), 401–439 (2009)CrossRefzbMATHMathSciNetGoogle Scholar - 11.Restrepo, R., Shin, J., Tetali, P., Vigoda, E., Yang, L.: Improved mixing condition on the grid for counting and sampling independent sets. Probab. Theory Relat. Fields
**156**, 75–99 (2013)CrossRefzbMATHMathSciNetGoogle Scholar - 12.Sly, A.: Computational transition at the uniqueness threshold. Proceedings of 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 287–296. IEEE Computer Society, Washington, DC (2010)Google Scholar
- 13.Sly, A., Sun, N.: The computational hardness of counting in two-spin models on \(d\)-regular graphs. Proceedings of 53rd Annual IEEE Symposium on the Foundations of Computer Science (FOCS), pp. 361–369. IEEE Computer Society, Washington, DC (2012)Google Scholar
- 14.Weitz, D.: Private, communication (2011)Google Scholar
- 15.Weitz, D.: Counting independent sets up to the tree threshold. Proceedings of 38th Annual ACM Symposium on Theory of Computing (STOC), pp. 140–149. ACM, New York (2006)Google Scholar
- 16.Zhang, J., Liang, H., Bai, F.: Approximating partition functions of the two-state spin system. Inf. Process. Lett.
**111**(14), 702–710 (2011)CrossRefzbMATHMathSciNetGoogle Scholar