skip to main content
research-article

Fast Algorithms for General Spin Systems on Bipartite Expanders

Published:01 September 2021Publication History
Skip Abstract Section

Abstract

A spin system is a framework in which the vertices of a graph are assigned spins from a finite set. The interactions between neighbouring spins give rise to weights, so a spin assignment can also be viewed as a weighted graph homomorphism. The problem of approximating the partition function (the aggregate weight of spin assignments) or of sampling from the resulting probability distribution is typically intractable for general graphs.

In this work, we consider arbitrary spin systems on bipartite expander Δ-regular graphs, including the canonical class of bipartite random Δ-regular graphs. We develop fast approximate sampling and counting algorithms for general spin systems whenever the degree and the spectral gap of the graph are sufficiently large. Roughly, this guarantees that the spin system is in the so-called low-temperature regime. Our approach generalises the techniques of Jenssen et al. and Chen et al. by showing that typical configurations on bipartite expanders correspond to “bicliques” of the spin system; then, using suitable polymer models, we show how to sample such configurations and approximate the partition function in Õ(n2) time, where n is the size of the graph.

References

  1. G. Brito, I. Dumitriu, and K. D. Harris. 2018. Spectral gap in random bipartite biregular graphs and applications. arXiv:1804.07808.Google ScholarGoogle Scholar
  2. A. Bulatov and M. Grohe. 2005. The complexity of partition functions. Theoretical Computer Science 348, 2 (2005), 148–186.Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. J.-Y. Cai, A. Galanis, L. A. Goldberg, H. Guo, M. Jerrum, D. Štefankovič, and E. Vigoda. 2016. #BIS-hardness for 2-spin systems on bipartite bounded degree graphs in the tree non-uniqueness region. Journal of Computer and System Sciences 82, 5 (2016), 690–711.Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. Sarah Cannon and Will Perkins. 2020. Counting independent sets in unbalanced bipartite graphs. In Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms. 1456–1466.Google ScholarGoogle ScholarCross RefCross Ref
  5. C. Carlson, E. Davies, and A. Kolla. 2020. Efficient algorithms for the Potts model on small-set expanders. arXiv:2003.01154.Google ScholarGoogle Scholar
  6. Zongchen Chen, Andreas Galanis, Leslie Ann Goldberg, Will Perkins, James Stewart, and Eric Vigoda. 2021. Fast algorithms at low temperatures via Markov chains. Random Structures & Algorithms 58, 2 (2021), 294–321. DOI:https://doi.org/10.1002/rsa.20968 Theorems 5 and 6 slightly updated after publication at https://arxiv.org/abs/1901.06653.Google ScholarGoogle ScholarCross RefCross Ref
  7. S. De Winter, J. Schillewaert, and J. Verstraete. 2012. Large incidence-free sets in geometries. Electronic Journal of Combinatorics 19, 4 (2012), P24.Google ScholarGoogle ScholarCross RefCross Ref
  8. M. Dyer and C. Greenhill. 2000. The complexity of counting graph homomorphisms. Random Structures & Algorithms 17, 3–4 (2000), 260–289.Google ScholarGoogle ScholarCross RefCross Ref
  9. M. E. Dyer, L. A. Goldberg, C. S. Greenhill, and M. Jerrum. 2004. The relative complexity of approximate counting problems. Algorithmica 38, 3 (2004), 471–500.Google ScholarGoogle ScholarCross RefCross Ref
  10. A. Galanis, L. A. Goldberg, and M. Jerrum. 2016. Approximately counting -colorings is #BIS-Hard. SIAM Journal on Computing 45, 3 (2016), 680–711.Google ScholarGoogle ScholarCross RefCross Ref
  11. A. Galanis, D. Štefankovič, and E. Vigoda. 2015. Inapproximability for antiferromagnetic spin systems in the tree nonuniqueness region. Journal of the ACM 62, 6 (2015), Article 50.Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. A. Galanis, D. Štefankovič, E. Vigoda, and L. Yang. 2016. Ferromagnetic Potts model: Refined #BIS-hardness and related results. SIAM Journal on Computing 45, 6 (2016), 2004–2065.Google ScholarGoogle ScholarCross RefCross Ref
  13. L. A. Goldberg and M. Jerrum. 2012. Approximating the partition function of the ferromagnetic Potts model. Journal of the ACM 59, 5 (2012), Article 25.Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Leslie Ann Goldberg and Mark Jerrum. 2014. The complexity of approximately counting tree homomorphisms. ACM Transactions on Computation Theory 6, 2 (2014), Article 8, 31 pages. DOI:https://doi.org/10.1145/2600917Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. L. A. Goldberg, S. Kelk, and M. Paterson. 2004. The complexity of choosing an -coloring (nearly) uniformly at random. SIAM Journal on Computing 33, 2 (2004), 416–432.Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. A. Govorov, J.-Y. Cai, and M. Dyer. 2020. A dichotomy for bounded degree graph homomorphisms with nonnegative weights. arXiv:2002.02021.Google ScholarGoogle Scholar
  17. C. Gruber and H. Kunz. 1971. General properties of polymer systems. Communications in Mathematical Physics 22, 2 (1971), 133–161.Google ScholarGoogle ScholarCross RefCross Ref
  18. W. H. Haemers. 1995. Interlacing eigenvalues and graphs. Linear Algebra and Its Applications 226, 228 (1995), 593–616.Google ScholarGoogle ScholarCross RefCross Ref
  19. Tyler Helmuth, Will Perkins, and Guus Regts. 2019. Algorithmic Pirogov–Sinai theory. Probability Theory and Related Fields 176 (2019), 851–895.Google ScholarGoogle ScholarCross RefCross Ref
  20. S. Hoory, N. Linial, and A. Wigderson. 2006. Expander graphs and their applications. Bulletin of the American Mathematical Society 43, 4 (2006), 439–561.Google ScholarGoogle ScholarCross RefCross Ref
  21. M. Jenssen, P. Keevash, and W. Perkins. 2019. Algorithms for #BIS-hard problems on expander graphs. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’19). 2235–2247.Google ScholarGoogle Scholar
  22. N. Kahale. 1995. Eigenvalues and expansion of regular graphs. Journal of the ACM 42, 5 (1995), 1091–1106. DOI:https://doi.org/10.1145/210118.210136Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. R. Kotecký and D. Preiss. 1986. Cluster expansion for abstract polymer models. Communications in Mathematical Physics 103, 3 (1986), 491–498.Google ScholarGoogle ScholarCross RefCross Ref
  24. C. Liao, J. Lin, P. Lu, and Z. Mao. 2019. Counting independent sets and colorings on random regular bipartite graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM’19). Leibniz International Proceedings in Informatics, Vol. 145. Article 34, 12 pages.Google ScholarGoogle Scholar
  25. A. Sly and N. Sun. 2014. Counting in two-spin models on -regular graphs. Annals of Probabability 42, 6 (11 2014), 2383–2416.Google ScholarGoogle Scholar
  26. R. M. Tanner. 1984. Explicit concentrators from generalized -gons. SIAM Journal on Algebraic Discrete Methods 5, 3 (1984), 287–293.Google ScholarGoogle ScholarCross RefCross Ref

Index Terms

  1. Fast Algorithms for General Spin Systems on Bipartite Expanders

    Recommendations

    Comments

    Login options

    Check if you have access through your login credentials or your institution to get full access on this article.

    Sign in

    Full Access

    • Published in

      cover image ACM Transactions on Computation Theory
      ACM Transactions on Computation Theory  Volume 13, Issue 4
      December 2021
      198 pages
      ISSN:1942-3454
      EISSN:1942-3462
      DOI:10.1145/3481683
      Issue’s Table of Contents

      Copyright © 2021 Association for Computing Machinery.

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      • Published: 1 September 2021
      • Accepted: 1 May 2021
      • Revised: 1 April 2021
      • Received: 1 July 2020
      Published in toct Volume 13, Issue 4

      Permissions

      Request permissions about this article.

      Request Permissions

      Check for updates

      Qualifiers

      • research-article
      • Refereed

    PDF Format

    View or Download as a PDF file.

    PDF

    eReader

    View online with eReader.

    eReader

    HTML Format

    View this article in HTML Format .

    View HTML Format
    About Cookies On This Site

    We use cookies to ensure that we give you the best experience on our website.

    Learn more

    Got it!