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The Nature of Computation$
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Cristopher Moore and Stephan Mertens

Print publication date: 2011

Print ISBN-13: 9780199233212

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199233212.001.0001

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Counting, Sampling, and Statistical Physics

Counting, Sampling, and Statistical Physics

Chapter:
(p.651) Chapter 13 Counting, Sampling, and Statistical Physics
Source:
The Nature of Computation
Author(s):

Cristopher Moore

Stephan Mertens

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199233212.003.0013

The objects that are solutions to an NP-complete problem are difficult to count. Counting can be a subtle and complex problem even when the corresponding existence and optimisation problems are in P. Spanning trees and perfect matchings are simple graph-theoretic objects, and the difference between them has deep mathematical roots. A matrix's determinant is the number of spanning trees while its permanent is the number of perfect matchings. Counting is closely associated with sampling. This chapter explores how to generate random matchings, and hence count them approximately, using a Markov chain that mixes in polynomial time. It considers the special case of planar graphs, such as the square lattice, to demonstrate that the number of perfect matchings is in P. It also discusses the implications of this fact for statistical physics and looks at how to find exact solutions for many physical models in two dimensions, including the Ising model.

Keywords:   spanning trees, perfect matchings, counting, sampling, random matchings, Markov chain, polynomial time, planar graphs, statistical physics, Ising model

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