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Approximate Counting for Spin Systems in Sub-Quadratic Time

Authors Konrad Anand , Weiming Feng , Graham Freifeld, Heng Guo , Jiaheng Wang

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ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Randomised algorithm
  • Approximate counting
  • Spin system
  • Sub-quadratic algorithm

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Abstract

We present two randomised approximate counting algorithms with Õ(n^{2-c}/ε²) running time for some constant c > 0 and accuracy ε:  
1) for the hard-core model with fugacity λ on graphs with maximum degree Δ when λ = O(Δ^{-1.5-c₁}) where c₁ = c/(2-2c); 
2) for spin systems with strong spatial mixing (SSM) on planar graphs with quadratic growth, such as ℤ². 
For the hard-core model, Weitz’s algorithm (STOC, 2006) achieves sub-quadratic running time when correlation decays faster than the neighbourhood growth, namely when λ = o(Δ^{-2}). Our first algorithm does not require this property and extends the range where sub-quadratic algorithms exist.
Our second algorithm appears to be the first to achieve sub-quadratic running time up to the SSM threshold, albeit on a restricted family of graphs. It also extends to (not necessarily planar) graphs with polynomial growth, such as ℤ^d, but with a running time of the form Õ(n²ε^{-2}/2^{c(log n)^{1/d}}) where d is the exponent of the polynomial growth and c > 0 is some constant.

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Konrad Anand, Weiming Feng, Graham Freifeld, Heng Guo, and Jiaheng Wang. Approximate Counting for Spin Systems in Sub-Quadratic Time. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.11

Author Details

Konrad Anand
  • School of Mathematical Sciences, Queen Mary University of London, London, UK
Weiming Feng
  • Institute for Theoretical Studies, ETH Zürich, Switzerland
Graham Freifeld
  • School of Informatics, University of Edinburgh, UK
Heng Guo
  • School of Informatics, University of Edinburgh, UK
Jiaheng Wang
  • School of Informatics, University of Edinburgh, UK

Funding

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 947778). Weiming Feng acknowledges support of Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zürich Foundation.

Acknowledgements

We would like to thank Chunyang Wang for pointing out how to shave a factor of e from Lemma 24.

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