An analysis of a Monte Carlo algorithm for estimating the permanent
 Alan Frieze,
 Mark Jerrum
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Abstract
Karmarkar, Karp, Lipton, Lovász, and Luby proposed a Monte Carlo algorithm for approximating the permanent of a nonnegativen×n matrix, which is based on an easily computed, unbiased estimator. It is not difficult to construct 0,1matrices for which the variance of this estimator is very large, so that an exponential number of trials is necessary to obtain a reliable approximation that is within a constant factor of the correct value.
Nevertheless, the same authors conjectured that for a random 0,1matrix the variance of the estimator is typically small. The conjecture is shown to be true; indeed, for almost every 0,1matrixA, just O(nw(n)e ^{2}) trials suffice to obtain a reliable approximation to the permanent ofA within a factor 1±ɛ of the correct value. Here ω(n) is any function tending to infinity asn→∞. This result extends to random 0,1matrices with density at leastn ^{−1/2}ω(n).
It is also shown that polynomially many trials suffice to approximate the permanent of any dense 0,1matrix, i.e., one in which every row and columnsum is at least (1/2+α)n, for some constant α>0. The degree of the polynomial bounding the number of trials is a function of α, and increases as α→0.
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 Title
 An analysis of a Monte Carlo algorithm for estimating the permanent
 Journal

Combinatorica
Volume 15, Issue 1 , pp 6783
 Cover Date
 19950301
 DOI
 10.1007/BF01294460
 Print ISSN
 02099683
 Online ISSN
 14396912
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 68 Q 25
 Industry Sectors
 Authors

 Alan Frieze ^{(1)}
 Mark Jerrum ^{(2)}
 Author Affiliations

 1. Department of Mathematics, Carnegie Mellon University, 15213, Pittsburgh, PA, U.S.A.
 2. Department of Computer Science, University of Edinburgh, The King's Buildings, EH9 3JZ, Edinburgh, UK