Fast Algorithms for General Spin Systems on Bipartite Expanders
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.
bipartite expanders
approximate counting
spin systems
Theory of computation~Design and analysis of algorithms
Theory of computation~Randomness, geometry and discrete structures
Mathematics of computing~Discrete mathematics
37:1-37:14
Regular Paper
A full version with all proofs can be found at https://arxiv.org/abs/2004.13442. The theorem numbers here match those in the full version. Proof sketches here refer to lemmas in the full version without restating them in detail.
Andreas
Galanis
Andreas Galanis
Department of Computer Science, University of Oxford, UK
Leslie Ann
Goldberg
Leslie Ann Goldberg
Department of Computer Science, University of Oxford, UK
James
Stewart
James Stewart
Department of Computer Science, University of Oxford, UK
10.4230/LIPIcs.MFCS.2020.37
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Andreas Galanis, Leslie Ann Goldberg, and James Stewart
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