A polynomial time additive estimate of the permanent using Gaussian fields

12/20/2022

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by   Tantrik Mukerji, et al.

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We present a polynomial-time randomized algorithm for estimating the permanent of an arbitrary M × M real matrix A up to an additive error. We do this by viewing the permanent of A as the expectation of a product of a centered joint Gaussian random variables whose covariance matrix we call the Gaussian embedding of A. The algorithm outputs the empirical mean S_N of this product after sampling from this multivariate distribution N times. In particular, after sampling N samples, our algorithm runs in time O(MN) with failure probability P(|S_N-perm(A)| > t) ≤3^M/t^2Nα^2M for α≥A.