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Fast Algorithms for General Spin Systems on Bipartite Expanders

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. 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM Transactions on Computation Theory (TOCT) Association for Computing Machinery

Fast Algorithms for General Spin Systems on Bipartite Expanders

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Publisher
Association for Computing Machinery
Copyright
Copyright © 2021 Association for Computing Machinery.
ISSN
1942-3454
eISSN
1942-3462
DOI
10.1145/3470865
Publisher site
See Article on Publisher Site

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.

Journal

ACM Transactions on Computation Theory (TOCT)Association for Computing Machinery

Published: Aug 31, 2021

Keywords: Bipartite expanders

References