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Improved FPTAS for Multi-spin Systems

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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2013, RANDOM 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8096))

Abstract

We design deterministic fully polynomial-time approximation scheme (FPTAS) for computing the partition function for a class of multi-spin systems, extending the known approximable regime by an exponential scale. As a consequence, we have an FPTAS for the Potts models with inverse temperature β up to a critical threshold \(|\beta|=O(\frac{1}{\Delta})\) where Δ is the maximum degree, confirming a conjecture in [10]. We also give an improved FPTAS for a generalization of counting q-colorings, namely the counting list-colorings. As a consequence we have an FPTAS for counting q-colorings in graphs with maximum degree Δ when q ≥ αΔ + 1 for α greater than α * ≈ 2.58071. This is so far the best bound achieved by deterministic approximation algorithms for counting q-colorings. All these improvements are obtained by applying a potential analysis to the correlation decay on computation trees for multi-spin systems.

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References

  1. Bandyopadhyay, A., Gamarnik, D.: Counting without sampling: Asymptotics of the log-partition function for certain statistical physics models. Random Structures & Algorithms 33(4), 452–479 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bayati, M., Gamarnik, D., Katz, D., Nair, C., Tetali, P.: Simple deterministic approximation algorithms for counting matchings. In: Proceedings of STOC, pp. 122–127 (2007)

    Google Scholar 

  3. Cai, J.-Y., Chen, X.: A decidable dichotomy theorem on directed graph homomorphisms with non-negative weights. In: Proceedings FOCS, pp. 437–446 (2010)

    Google Scholar 

  4. Cai, J.-Y., Chen, X., Lu, P.: Graph homomorphisms with complex values: A dichotomy theorem. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 275–286. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  5. Dyer, M., Jerrum, M., Vigoda, E.: Rapidly mixing markov chains for dismantleable constraint graphs. In: Rolim, J.D.P., Vadhan, S.P. (eds.) RANDOM 2002. LNCS, vol. 2483, pp. 68–77. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  6. Dyer, M.E., Frieze, A.M., Hayes, T.P., Vigoda, E.: Randomly coloring constant degree graphs. In: Proceedings of FOCS, pp. 582–589 (2004)

    Google Scholar 

  7. Dyer, M.E., Greenhill, C.S.: On markov chains for independent sets. Journal of Algorithms 35(1), 17–49 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Galanis, A., Štefankovič, D., Vigoda, E.: Inapproximability of the partition function for the anti-ferromagnetic ising and hard-core models. arXiv preprint arXiv:1203.2226 (2012)

    Google Scholar 

  9. Galanis, A., Štefankovič, D., Vigoda, E.: Inapproximability for anti-ferromagnetic spin systems in the tree non-uniqueness region. arXiv preprint arXiv:1305.2902 (2013)

    Google Scholar 

  10. Gamarnik, D., Katz, D.: Correlation decay and deterministic FPTAS for counting colorings of a graph. Journal of Discrete Algorithms 12, 29–47 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gamarnik, D., Katz, D., Misra, S.: Strong spatial mixing for list coloring of graphs. arXiv preprint arXiv:1207.1223 (2012)

    Google Scholar 

  12. Goldberg, L.A., Martin, R., Paterson, M.: Strong spatial mixing with fewer colors for lattice graphs. SIAM Journal on Computing 35(2), 486 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Goldberg, L.A., Jerrum, M.: A polynomial-time algorithm for estimating the partition function of the ferromagnetic ising model on a regular matroid. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 521–532. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  14. Goldberg, L.A., Jerrum, M., Paterson, M.: The computational complexity of two-state spin systems. Random Structures & Algorithms 23(2), 133–154 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hayes, T.P.: Randomly coloring graphs of girth at least five. In: Proceedings of STOC, pp. 269–278 (2003)

    Google Scholar 

  16. Jerrum, M.: A very simple algorithm for estimating the number of k-colorings of a low-degree graph. Random Structures & Algorithms 7(2), 157–166 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jerrum, M., Sinclair, A.: Polynomial-time approximation algorithms for the ising model. SIAM Journal on Computing 22(5), 1087–1116 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. Journal of the ACM 51, 671–697 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, L., Lu, P., Yin, Y.: Approximate counting via correlation decay in spin systems. In: Proceedings of SODA, pp. 922–940 (2012)

    Google Scholar 

  20. Li, L., Lu, P., Yin, Y.: Correlation decay up to uniqueness in spin systems. In: Proceedings of SODA, pp. 67–84 (2013)

    Google Scholar 

  21. Luby, M., Vigoda, E.: Approximately counting up to four (extended abstract). In: Proceedings of STOC, pp. 682–687 (1997)

    Google Scholar 

  22. Restrepo, R., Shin, J., Tetali, P., Vigoda, E., Yang, L.: Improved mixing condition on the grid for counting and sampling independent sets. In: Proceedings of FOCS, pp. 140–149 (2011)

    Google Scholar 

  23. Sinclair, A., Srivastava, P., Thurley, M.: Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs. In: Proceedings of SODA, pp. 941–953 (2012)

    Google Scholar 

  24. Sly, A.: Uniqueness thresholds on trees versus graphs. The Annals of Applied Probability 18(5), 1897–1909 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sly, A.: Computational transition at the uniqueness threshold. In: Proceedings of FOCS, pp. 287–296 (2010)

    Google Scholar 

  26. Sly, A., Sun, N.: The computational hardness of counting in two-spin models on d-regular graphs. In: Proceedings of FOCS, pp. 361–369 (2012)

    Google Scholar 

  27. Vigoda, E.: Improved bounds for sampling coloring. In: Proceedings of FOCS, pp. 51–59 (1999)

    Google Scholar 

  28. Weitz, D.: Counting independent sets up to the tree threshold. In: Proceedings of STOC, pp. 140–149 (2006)

    Google Scholar 

  29. Yin, Y., Zhang, C.: Approximate counting via correlation decay on planar graphs. In: Proceedings of SODA, pp. 47–66 (2013)

    Google Scholar 

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Authors and Affiliations

  1. Microsoft Research Asia, China

    Pinyan Lu

  2. State Key Laboratory for Novel Software Technology, Nanjing University, China

    Yitong Yin

Authors
  1. Pinyan Lu
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  2. Yitong Yin
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Editor information

Editors and Affiliations

  1. University of California, Berkeley, CA, USA

    Prasad Raghavendra

  2. Dept. of computer Science and Engineering, Pennsylvania State University, University Park, PA, USA

    Sofya Raskhodnikova

  3. Institute of Computer Science, University of Kiel, 24118, Kiel, Germany

    Klaus Jansen

  4. University of Geneva, Centre Universitaire d’Informatique, 1227, Carouge, Switzerland

    José D. P. Rolim

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© 2013 Springer-Verlag Berlin Heidelberg

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Lu, P., Yin, Y. (2013). Improved FPTAS for Multi-spin Systems. In: Raghavendra, P., Raskhodnikova, S., Jansen, K., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2013 2013. Lecture Notes in Computer Science, vol 8096. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40328-6_44

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  • DOI: https://doi.org/10.1007/978-3-642-40328-6_44

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40327-9

  • Online ISBN: 978-3-642-40328-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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