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Polynomial-time approximation algorithms for the antiferromagnetic Ising model on line graphs

Part of: Equilibrium statistical mechanics Markov processes Algorithms - Computer Science Theory of computing

Published online by Cambridge University Press:  12 April 2021

Martin Dyer ,
Marc Heinrich ,
Mark Jerrum  and
Haiko Müller
Martin Dyer
Affiliation:
School of Computing, University of Leeds, Leeds LS2 9JT, UK
Marc Heinrich
Affiliation:
School of Computing, University of Leeds, Leeds LS2 9JT, UK
Mark Jerrum*
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, LondonE1 4NS, UK
Haiko Müller
Affiliation:
School of Computing, University of Leeds, Leeds LS2 9JT, UK
*
*Corressponding author. Email: m.jerrum@qmul.ac.uk
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Abstract

We present a polynomial-time Markov chain Monte Carlo algorithm for estimating the partition function of the antiferromagnetic Ising model on any line graph. The analysis of the algorithm exploits the ‘winding’ technology devised by McQuillan [CoRR abs/1301.2880 (2013)] and developed by Huang, Lu and Zhang [Proc. 27th Symp. on Disc. Algorithms (SODA16), 514–527]. We show that exact computation of the partition function is #P-hard, even for line graphs, indicating that an approximation algorithm is the best that can be expected. We also show that Glauber dynamics for the Ising model is rapidly mixing on line graphs, an example being the kagome lattice.

MSC classification

Primary: 68Q25: Analysis of algorithms and problem complexity
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 68Q17: Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) 68W20: Randomized algorithms 68W25: Approximation algorithms 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 6 , November 2021 , pp. 905 - 921
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

†

All authors are supported by EPSRC grants EP/S016562/1 and EP/S016694/1, ‘Sampling in hereditary classes’.

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