research-article

Fast Algorithms for General Spin Systems on Bipartite Expanders

Authors:

Andreas Galanis,

Leslie Ann Goldberg,

James StewartAuthors Info & Claims

ACM Transactions on Computation Theory (TOCT), Volume 13, Issue 4

Article No.: 25, Pages 1 - 18

https://doi.org/10.1145/3470865

Published: 01 September 2021 Publication History

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.

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Cited By

Blanca ACannon SPerkins W(2024)Fast and Perfect Sampling of Subgraphs and Polymer SystemsACM Transactions on Algorithms10.1145/363229420:1(1-30)Online publication date: 22-Jan-2024
https://dl.acm.org/doi/10.1145/3632294
Jenssen MPerkins WPotukuchi ASimkin M(2024)Sampling, Counting, and Large Deviations for Triangle-Free Graphs Near the Critical Density2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00020(151-165)Online publication date: 27-Oct-2024
https://doi.org/10.1109/FOCS61266.2024.00020
Carlson CDavies EFraiman NKolla APotukuchi AYap C(2024)Algorithms for the ferromagnetic Potts model on expandersCombinatorics, Probability and Computing10.1017/S0963548324000087(1-31)Online publication date: 5-Apr-2024
https://doi.org/10.1017/S0963548324000087
Show More Cited By

Index Terms

Fast Algorithms for General Spin Systems on Bipartite Expanders
1. Theory of computation
  1. Design and analysis of algorithms

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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

Editor:
Ryan O'Donnell
Carnegie Mellon University, USA

Issue’s Table of Contents

Copyright © 2021 Association for Computing Machinery.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 September 2021

Accepted: 01 May 2021

Revised: 01 April 2021

Received: 01 July 2020

Published in TOCT Volume 13, Issue 4

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Cited By

Blanca ACannon SPerkins W(2024)Fast and Perfect Sampling of Subgraphs and Polymer SystemsACM Transactions on Algorithms10.1145/363229420:1(1-30)Online publication date: 22-Jan-2024
https://dl.acm.org/doi/10.1145/3632294
Jenssen MPerkins WPotukuchi ASimkin M(2024)Sampling, Counting, and Large Deviations for Triangle-Free Graphs Near the Critical Density2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00020(151-165)Online publication date: 27-Oct-2024
https://doi.org/10.1109/FOCS61266.2024.00020
Carlson CDavies EFraiman NKolla APotukuchi AYap C(2024)Algorithms for the ferromagnetic Potts model on expandersCombinatorics, Probability and Computing10.1017/S0963548324000087(1-31)Online publication date: 5-Apr-2024
https://doi.org/10.1017/S0963548324000087
Jenssen M(2024)The Cluster Expansion in CombinatoricsSurveys in Combinatorics 202410.1017/9781009490559.004(55-88)Online publication date: 23-May-2024
https://doi.org/10.1017/9781009490559.004
Friedrich TGöbel AKrejca MPappik M(2023)Polymer dynamics via cliquesTheoretical Computer Science10.1016/j.tcs.2022.11.035942:C(230-252)Online publication date: 9-Jan-2023
https://dl.acm.org/doi/10.1016/j.tcs.2022.11.035
Jenssen MKeevash P(2023)Homomorphisms from the torusAdvances in Mathematics10.1016/j.aim.2023.109212430(109212)Online publication date: Oct-2023
https://doi.org/10.1016/j.aim.2023.109212
Carlson CDavies EFraiman NKolla APotukuchi AYap C(2022)Algorithms for the ferromagnetic Potts model on expanders2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00040(344-355)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00040

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