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Improved mixing condition on the grid for counting and sampling independent sets
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  • Published: 25 March 2012

Improved mixing condition on the grid for counting and sampling independent sets

  • Ricardo Restrepo1,
  • Jinwoo Shin2,
  • Prasad Tetali3,
  • Eric Vigoda2 &
  • …
  • Linji Yang2 

Probability Theory and Related Fields volume 156, pages 75–99 (2013)Cite this article

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Abstract

The hard-core model has received much attention in the past couple of decades as a lattice gas model with hard constraints in statistical physics, a multicast model of calls in communication networks, and as a weighted independent set problem in combinatorics, probability and theoretical computer science. In this model, each independent set I in a graph G is weighted proportionally to λ|I|, for a positive real parameter λ. For large λ, computing the partition function (namely, the normalizing constant which makes the weighting a probability distribution on a finite graph) on graphs of maximum degree Δ ≥ 3, is a well known computationally challenging problem. More concretely, let \({\lambda_c(\mathbb{T}_\Delta)}\) denote the critical value for the so-called uniqueness threshold of the hard-core model on the infinite Δ-regular tree; recent breakthrough results of Weitz (Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pp. 140–149, 2006) and Sly (Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 287–296, 2010) have identified \({\lambda_c(\mathbb{T}_\Delta)}\) as a threshold where the hardness of estimating the above partition function undergoes a computational transition. We focus on the well-studied particular case of the square lattice \({\mathbb{Z}^2}\) , and provide a new lower bound for the uniqueness threshold, in particular taking it well above \({\lambda_c(\mathbb{T}_4)}\) . Our technique refines and builds on the tree of self-avoiding walks approach of Weitz, resulting in a new technical sufficient criterion (of wider applicability) for establishing strong spatial mixing (and hence uniqueness) for the hard-core model. Our new criterion achieves better bounds on strong spatial mixing when the graph has extra structure, improving upon what can be achieved by just using the maximum degree. Applying our technique to \({\mathbb{Z}^2}\) we prove that strong spatial mixing holds for all λ < 2.3882, improving upon the work of Weitz that held for λ < 27/16 = 1.6875. Our results imply a fully-polynomial deterministic approximation algorithm for estimating the partition function, as well as rapid mixing of the associated Glauber dynamics to sample from the hard-core distribution.

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References

  1. Alm S.E.: Upper bounds for the connective constant of self-avoiding walks. Combin. Probab. Comput. 4(2), 115–136 (1993)

    MathSciNet  Google Scholar 

  2. Arnon, D.S., Collins, G.E., McCallum, S.: Cylindrical algebraic decomposition I: the basic algorithm. SIAM J. Comput. 865–877 (1984)

  3. Baxter R.J., Enting I.G., Tsang S.K.: Hard-square lattice gas. J. Stat. Phys. 22(4), 465–489 (1980)

    Article  MathSciNet  Google Scholar 

  4. Beffara, V., Duminil-Copin, H.: The self-dual point of the two-dimensional random cluster model is critical for q ≥ 1. Probab. Theory Relat. Fields. Preprint available from the arXiv at: http://arxiv.org/abs/1006.5073

  5. van den Berg J., Ermakov A.: A new lower bound for the critical probability of site percolation on the square lattice. Random Struct. Algorithms 8(3), 199–212 (1996)

    Article  MATH  Google Scholar 

  6. van den Berg J., Steif J.E.: Percolation and the hard-core lattice gas model. Stochastic Processes Appl. 49(2), 179–197 (1994)

    Article  MATH  Google Scholar 

  7. Brightwell G.R., Häggström O., Winkler P.: Nonmonotonic behavior in hard-core and Widom–Rowlinson models. J. Stat. Phys. 94(3), 415–435 (1999)

    Article  MATH  Google Scholar 

  8. Cesi F.: Quasi–factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields. Probab. Theory Relat. Fields 120(4), 569–584 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dobrushin R.L.: The problem of uniqueness of a Gibbsian random field and the problem of phase transitions. Funct. Anal. Appl. 2(4), 302–312 (1968)

    Article  MATH  Google Scholar 

  10. Dobrushin R.L., Shlosman S.B.: Constructive unicity criterion. In: Fritz, J., Jaffe, A., Szasz, D. (eds) Statistical Mechanics and Dynamical Systems, pp. 347–370. Birkhäuser, New York (1985)

    Google Scholar 

  11. Dyer M.E., Sinclair A., Vigoda E., Weitz D.: Mixing in time and space for lattice spin systems: a combinatorial view. Random Struct. Algorithms 24(4), 461–479 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Galanis, A., Ge, Q., Štefankovič, D., Vigoda, E., Yang, L.: Improved inapproximability results for counting independent sets in the hard-core model. In: Proocedings of the 15th International Workshop, RANDOM, pp. 567–578 (2011)

  13. Gaunt D.S., Fisher M.E.: Hard-sphere lattice gases. I. Plane-square lattice. J. Chem. Phys. 43(8), 2840–2863 (1965)

    Article  MathSciNet  Google Scholar 

  14. Georgii H.-O.: Gibbs Measures and Phase Transitions. de Gruyter, Berlin (1988)

    Book  MATH  Google Scholar 

  15. Goldberg L., Martin R., Paterson M.: Strong spatial mixing for lattice graphs with fewer colours. SIAM J. Comput. 35(2), 486–517 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Greenhill C.: The complexity of counting colourings and independent sets in sparse graphs and hypergraphs. Comput. Complex. 9(1), 52–72 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jerrum M.R., Valiant L.G., Vazirani V.V.: Random generation of combinatorial structures from a uniform distribution distribution. Theor. Comput. Sci. 43(2-3), 169–186 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kelly F.P.: Loss networks. Ann. Appl. Probab. 1(3), 319–378 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  19. Levin D.A., Peres Y., Wilmer E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2008)

    Google Scholar 

  20. Lyons R.: The Ising model and percolation on trees and tree-like graphs. Commun. Math. Phys. 125(2), 337–353 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  21. Martinelli, F.: Lectures on Glauber dynamics for discrete spin models. In: Lectures on Probability Theory and Statistics (Saint-Flour, 1997), Lecture Notes in Mathematics, vol. 1717, pp. 93–191 (1998)

  22. Martinelli F., Olivieri E.: Approach to equilibrium of Glauber dynamics in the one phase region. I. The attractive case. Commun. Math. Phys. 161(3), 447–486 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Martinelli F., Olivieri E.: Approach to equilibrium of Glauber dynamics in the one phase region. II. The general case. Commun. Math. Phys. 161(3), 487–514 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Onsager L.: Crystal statistics. I. A two-dimensional model with an order–disorder transition. Phys. Rev. Lett. 65(3–4), 117–149 (1944)

    MathSciNet  MATH  Google Scholar 

  25. Pönitz A., Tittman P.: Improved upper bounds for self-avoiding walks in \({\mathbb{Z}^d}\) . Electr. J. Combin. 7(1), R21 (2000)

    Google Scholar 

  26. Rácz Z.: Phase boundary of Ising antiferromagnets near H = H c and T = 0: results from hard-core lattice gas calculations. Phys. Rev. B 21(9), 4012–4016 (1980)

    Article  Google Scholar 

  27. Radulescu, D.C.: A computer-assisted proof of uniqueness of phase for the hard-square lattice gas model in two dimensions. PhD dissertation, Rutgers University, New Brunswick, NJ, USA (1997)

  28. Radulescu D.C., Styer D.F.: The Dobrushin–Shlosman phase uniqueness criterion and applications to hard squares. J. Stat. Phys. 49(1–2), 281–295 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  29. Randall, D.: Slow mixing of Glauber dynamics via topological obstructions. In: Proceedings of the 17th Annual ACM–SIAM Symposium on Discrete Algorithms (SODA), pp. 870–879 (2006)

  30. Sly, A.: Computational transition at the uniqueness threshold. In: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 287–296 (2010)

  31. Štefankovič D., Vempala S., Vigoda E.: Adaptive simulated annealing: a near-optimal connection between sampling and counting. J. ACM 56(3), 1–36 (2009)

    MathSciNet  Google Scholar 

  32. Tarski A.: A decision method for elementary algebra and geometry, 2nd edn. University of California Press, California (1951)

    Google Scholar 

  33. Valiant L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8(3), 410–421 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  34. Weitz, D.: Counting independent sets up to the tree threshold. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pp. 140–149 (2006)

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Authors and Affiliations

  1. Universidad de Antioquia, Medellin, Colombia

    Ricardo Restrepo

  2. School of Computer Science, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Jinwoo Shin, Eric Vigoda & Linji Yang

  3. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Prasad Tetali

Authors
  1. Ricardo Restrepo
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  2. Jinwoo Shin
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  3. Prasad Tetali
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  4. Eric Vigoda
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  5. Linji Yang
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Corresponding author

Correspondence to Eric Vigoda.

Additional information

Supported by NSF grants CCF-0830298 and CCF-0910584. Jinwoo Shin was supported by the Algorithms and Randomness Center at Georgia Technology.

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Restrepo, R., Shin, J., Tetali, P. et al. Improved mixing condition on the grid for counting and sampling independent sets. Probab. Theory Relat. Fields 156, 75–99 (2013). https://doi.org/10.1007/s00440-012-0421-8

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  • Received: 14 June 2011

  • Revised: 27 February 2012

  • Accepted: 29 February 2012

  • Published: 25 March 2012

  • Issue Date: June 2013

  • DOI: https://doi.org/10.1007/s00440-012-0421-8

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Keywords

  • Lattice gas
  • Gibbs measures
  • Phase transition
  • Approximation algorithm
  • Glauber dynamics

Mathematics Subject Classification

  • 82B20
  • 68Q25
  • 60J10
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