NDLI: Approximating the Permanent

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Sinclair, Alistair Jerrum, Mark
Copyright Year	1989
Abstract	A randomised approximation scheme for the permanent of a 01s presented. The task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph; the latter is accomplished by simulating a Markov chain whose states are the matchings in the graph. For a wide class of 01 matrices the approximation scheme is fully-polynomial, i.e., runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output. This class includes all dense matrices (those that contain sufficiently many 1s) and almost all sparse matrices in some reasonable probabilistic model for 01 matrices of given density.For the approach sketched above to be computationally efficient, the Markov chain must be rapidly mixing: informally, it must converge in a short time to its stationary distribution. A major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing, apparently the first such result for a Markov chain with genuinely complex structure. The techniques used seem to have general applicability, and are applied again in the paper to validate a fully-polynomial randomised approximation scheme for the partition function of an arbitrary monomer-dimer system.
Starting Page	1149
Ending Page	1178
Page Count	30
File Format	PDF
ISSN	00975397
DOI	10.1137/0218077
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	6
Volume Number	18
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-13
Access Restriction	Subscribed
Subject Keyword	Applications of Markov chains and discrete-time Markov processes on general state spaces statistical physics Factorization, matching, partitioning, covering and packing random generation simulated annealing perfect matchings Random graphs permanent monomer-dimer systems Markov chains rapid mixing counting problems
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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