An analysis of a Monte Carlo algorithm for estimating the permanent

Abstract

Karmarkar, Karp, Lipton, Lovász, and Luby proposed a Monte Carlo algorithm for approximating the permanent of a non-negativen×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-matrixA, 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 0,1-matrices with density at leastn −1/2ω(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.

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Supported by NSF grant CCR-9225008.

The work described here was partly carried out while the author was visiting Princeton University as a guest of DIMACS (Center for Discrete Mathematics and Computer Science).

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Frieze, A., Jerrum, M. An analysis of a Monte Carlo algorithm for estimating the permanent. Combinatorica 15, 67–83 (1995). https://doi.org/10.1007/BF01294460

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Mathematics Subject Classification (1991)

  • 68 Q 25