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Sublinear-Time Algorithms for Monomer-Dimer Systems on Bounded Degree Graphs

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Algorithms and Computation (ISAAC 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8283))

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Abstract

For a graph G, let Z(G,λ) be the partition function of the monomer-dimer system defined by ∑  k m k (G)λ k, where m k (G) is the number of matchings of size k in G. We consider graphs of bounded degree and develop a sublinear-time algorithm for estimating logZ(G,λ) at an arbitrary value λ > 0 within additive error εn with high probability. The query complexity of our algorithm does not depend on the size of G and is polynomial in 1/ε, and we also provide a lower bound quadratic in 1/ε for this problem. This is the first analysis of a sublinear-time approximation algorithm for a # P-complete problem. Our approach is based on the correlation decay of the Gibbs distribution associated with Z(G,λ). We show that our algorithm approximates the probability for a vertex to be covered by a matching, sampled according to this Gibbs distribution, in a near-optimal sublinear time. We extend our results to approximate the average size and the entropy of such a matching within an additive error with high probability, where again the query complexity is polynomial in 1/ε and the lower bound is quadratic in 1/ε. Our algorithms are simple to implement and of practical use when dealing with massive datasets. Our results extend to other systems where the correlation decay is known to hold as for the independent set problem up to the critical activity.

Full version available at http://arxiv.org/abs/1208.3629

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References

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

    Google Scholar 

  2. Blume, L.E.: The statistical mechanics of strategic interaction. Games Econom. Behav. 5(3), 387–424 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bordenave, C., Lelarge, M., Salez, J.: Matchings on infinite graphs. Probability Theory and Related Fields 157(1-2), 183–208 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Canetti, R., Even, G., Goldreich, O.: Lower bounds for sampling algorithms for estimating the average. Information Processing Letters 53(1), 17–25 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chazelle, B., Rubinfeld, R., Trevisan, L.: Approximating the minimum spanning tree weight in sublinear time. SIAM Journal on computing 34(6), 1370–1379 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Czumaj, A., Sohler, C.: Sublinear-time algorithms. Bulletin of the EATCS 89, 23–47 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Feige, U.: On sums of independent random variables with unbounded variance and estimating the average degree in a graph. SIAM Journal on Computing 35(4), 964–984 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gamarnik, D., Katz, D.: A deterministic approximation algorithm for computing the permanent of a 0, 1 matrix. Journal of Computer and System Sciences 76(8), 879–883 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gamarnik, D., Katz, D.: Correlation decay and deterministic FPTAS for counting list-colorings of a graph. In: SODA, pp. 1245–1254. SIAM (2007)

    Google Scholar 

  10. Godsil, C.D.: Matchings and walks in graphs. J. Graph Theory 5(3), 285–297 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  11. Goldreich, O., Ron, D.: Approximating average parameters of graphs. Random Structures and Algorithms 32(4), 473–493 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gonen, M., Ron, D., Shavitt, Y.: Counting stars and other small subgraphs in sublinear-time. SIAM Journal on Discrete Mathematics 25(3), 1365–1411 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Heilmann, O.J., Lieb, E.H.: Theory of monomer-dimer systems. Comm. Math. Phys. 25, 190–232 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jerrum, M.: Counting, sampling and integrating: algorithms and complexity. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel (2003)

    Google Scholar 

  15. 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(4), 671–697 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jerrum, M., Valiant, L., Vazirani, V.: Random generation of combinatorial structures from a uniform distribution. Theoret. Comput. Sci. 43(2-3), 169–188 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  19. Lovász, L., Plummer, M.D.: Matching theory. AMS Chelsea Publishing, Providence (2009) corrected reprint of the 1986 original (MR0859549)

    Google Scholar 

  20. Mézard, M., Montanari, A.: Information, physics, and computation. Oxford Graduate Texts. Oxford University Press, Oxford (2009)

    Book  MATH  Google Scholar 

  21. Nguyen, H., Onak, K.: Constant-time approximation algorithms via local improvements. In: FOCS, pp. 327–336. IEEE (2008)

    Google Scholar 

  22. Onak, K., Ron, D., Rosen, M., Rubinfeld, R.: A near-optimal sublinear-time algorithm for approximating the minimum vertex cover size. In: SODA, pp. 1123–1131. SIAM (2012)

    Google Scholar 

  23. Parnas, M., Ron, D.: Approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms. Theoret. Comput. Sci. 381(1-3), 183–196 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rubinfeld, R.: Sublinear time algorithms. In: International Congress of Mathematicians, vol. III, pp. 1095–1110. Eur. Math. Soc., Zürich (2006)

    Google Scholar 

  25. Sinclair, A.: Algorithms for random generation and counting. In: Progress in Theoretical Computer Science, Birkhäuser Boston Inc., Boston (1993)

    Google Scholar 

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

    Google Scholar 

  27. Sinclair, A., Srivastava, P.: Lee-yang theorems and the complexity of computing averages. In: STOC, pp. 625–634. ACM (2013)

    Google Scholar 

  28. Vadhan, S.P.: The complexity of counting in sparse, regular, and planar graphs. SIAM Journal on Computing 31(2), 398–427 (2002)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  30. Welsh, D.J.A.: Complexity: knots, colourings and counting. London Mathematical Society Lecture Note Series, vol. 186. Cambridge University Press (1993)

    Google Scholar 

  31. Yao, A.: Probabilistic computations: Toward a unified measure of complexity. In: FOCS, pp. 222–227. IEEE (1977)

    Google Scholar 

  32. Yoshida, Y., Yamamoto, M., Ito, H.: An improved constant-time approximation algorithm for maximum independent sets and maximum matchings. In: STOC, pp. 225–234. ACM (2009)

    Google Scholar 

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Author information

Authors and Affiliations

  1. INRIA, École Normale Supérieure, France

    Marc Lelarge & Hang Zhou

Authors
  1. Marc Lelarge
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  2. Hang Zhou
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Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, Chinese University of Hong Kong, Shatin, 100084, Hong Kong, China

    Leizhen Cai

  2. Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Clear Water Bay, 100044, Hong Kong, China

    Siu-Wing Cheng

  3. Department of Computer Science, University of Hong Kong, Porfkulam, 100084, Hong Kong, China

    Tak-Wah Lam

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Lelarge, M., Zhou, H. (2013). Sublinear-Time Algorithms for Monomer-Dimer Systems on Bounded Degree Graphs. In: Cai, L., Cheng, SW., Lam, TW. (eds) Algorithms and Computation. ISAAC 2013. Lecture Notes in Computer Science, vol 8283. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45030-3_14

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  • DOI: https://doi.org/10.1007/978-3-642-45030-3_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-45029-7

  • Online ISBN: 978-3-642-45030-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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