Affiliation:
1. University of Oxford, Oxford, UK
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(
n
2
) time, where
n
is the size of the graph.
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Theory and Mathematics,Theoretical Computer Science
Reference26 articles.
1. G. Brito I. Dumitriu and K. D. Harris. 2018. Spectral gap in random bipartite biregular graphs and applications. arXiv:1804.07808. G. Brito I. Dumitriu and K. D. Harris. 2018. Spectral gap in random bipartite biregular graphs and applications. arXiv:1804.07808.
2. The complexity of partition functions
3. #BIS-hardness for 2-spin systems on bipartite bounded degree graphs in the tree non-uniqueness region
4. C. Carlson E. Davies and A. Kolla. 2020. Efficient algorithms for the Potts model on small-set expanders. arXiv:2003.01154. C. Carlson E. Davies and A. Kolla. 2020. Efficient algorithms for the Potts model on small-set expanders. arXiv:2003.01154.
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