Fast Algorithms for General Spin Systems on Bipartite Expanders

Authors Andreas Galanis, Leslie Ann Goldberg, James Stewart

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Author Details

Andreas Galanis
  • Department of Computer Science, University of Oxford, UK
Leslie Ann Goldberg
  • Department of Computer Science, University of Oxford, UK
James Stewart
  • Department of Computer Science, University of Oxford, UK

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Andreas Galanis, Leslie Ann Goldberg, and James Stewart. Fast Algorithms for General Spin Systems on Bipartite Expanders. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 37:1-37:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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. 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Randomness, geometry and discrete structures
  • Mathematics of computing → Discrete mathematics
  • bipartite expanders
  • approximate counting
  • spin systems


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