Fast Algorithms at Low Temperatures via Markov Chains
For spin systems, such as the hard-core model on independent sets weighted by fugacity lambda>0, efficient algorithms for the associated approximate counting/sampling problems typically apply in the high-temperature region, corresponding to low fugacity. Recent work of Jenssen, Keevash and Perkins (2019) yields an FPTAS for approximating the partition function (and an efficient sampling algorithm) on bounded-degree (bipartite) expander graphs for the hard-core model at sufficiently high fugacity, and also the ferromagnetic Potts model at sufficiently low temperatures. Their method is based on using the cluster expansion to obtain a complex zero-free region for the partition function of a polymer model, and then approximating this partition function using the polynomial interpolation method of Barvinok. We present a simple discrete-time Markov chain for abstract polymer models, and present an elementary proof of rapid mixing of this new chain under sufficient decay of the polymer weights. Applying these general polymer results to the hard-core and ferromagnetic Potts models on bounded-degree (bipartite) expander graphs yields fast algorithms with running time O(n log n) for the Potts model and O(n^2 log n) for the hard-core model, in contrast to typical running times of n^{O(log Delta)} for algorithms based on Barvinok’s polynomial interpolation method on graphs of maximum degree Delta. In addition, our approach via our polymer model Markov chain is conceptually simpler as it circumvents the zero-free analysis and the generalization to complex parameters. Finally, we combine our results for the hard-core and ferromagnetic Potts models with standard Markov chain comparison tools to obtain polynomial mixing time for the usual spin system Glauber dynamics restricted to even and odd or "red" dominant portions of the respective state spaces.
Markov chains
approximate counting
Potts model
hard-core model
expander graphs
Theory of computation~Random walks and Markov chains
Theory of computation~Design and analysis of algorithms
41:1-41:14
RANDOM
A full version of the paper is available at https://arxiv.org/abs/1901.06653, and the theorem numbering here matches that of the full version.
Zongchen
Chen
Zongchen Chen
School of Computer Science, Georgia Institute of Technology, Atlanta, USA
Research supported in part by NSF grants CCF-1617306 and CCF-1563838.
Andreas
Galanis
Andreas Galanis
Department of Computer Science, University of Oxford, Oxford, UK
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) ERC grant agreement no. 334828. The paper reflects only the authors' views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein.
Leslie Ann
Goldberg
Leslie Ann Goldberg
Department of Computer Science, University of Oxford, Oxford, UK
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) ERC grant agreement no. 334828. The paper reflects only the authors' views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein.
Will
Perkins
Will Perkins
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, USA
Part of this work was done while WP was visiting the Simons Institute for the Theory of Computing.
James
Stewart
James Stewart
Department of Computer Science, University of Oxford, Oxford, UK
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) ERC grant agreement no. 334828. The paper reflects only the authors' views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein.
Eric
Vigoda
Eric Vigoda
School of Computer Science, Georgia Institute of Technology, Atlanta, USA
Research supported in part by NSF grants CCF-1617306 and CCF-1563838.
10.4230/LIPIcs.APPROX-RANDOM.2019.41
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Zongchen Chen, Andreas Galanis, Leslie Ann Goldberg, Will Perkins, James Stewart, and Eric Vigoda
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