# An analysis of a Monte Carlo algorithm for estimating the permanent

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## Abstract

Karmarkar, Karp, Lipton, Lovász, and Luby proposed a Monte Carlo algorithm for approximating the permanent of a non-negative*n×n* matrix, which is based on an easily computed, unbiased estimator. It is not difficult to construct 0,1-matrices 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 0,1-matrix the variance of the estimator is typically small. The conjecture is shown to be true; indeed, for almost every 0,1-matrix*A*, just O(*nw(n)e*^{-2}) trials suffice to obtain a reliable approximation to the permanent of*A* within a factor 1±ɛ of the correct value. Here ω(*n*) is any function tending to infinity as*n*→∞. This result extends to random 0,1-matrices with density at least*n*^{−1/2}ω(*n*).

It is also shown that polynomially many trials suffice to approximate the permanent of any dense 0,1-matrix, i.e., one in which every row- and column-sum 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.

### Mathematics Subject Classification (1991)

68 Q 25## Preview

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