Additive estimates of the permanent using Gaussian fields

TL;DR

This paper introduces a randomized Gaussian-based algorithm for estimating the permanent of a matrix with additive error, providing a new approach that relates to semidefinite programming and combinatorial optimization.

Contribution

The paper presents a novel Gaussian field-based method for permanent estimation, improving computational efficiency and connecting to semidefinite programming and Max-Cut problems.

Findings

Algorithm runs in polynomial time with explicit error bounds.

Estimates the permanent via expectation of Gaussian products.

Links to semidefinite programming and combinatorial optimization.

Abstract

We present a randomized algorithm for estimating the permanent of an $M \times M$ real matrix $A$ up to an additive error. We do this by viewing the permanent $\mathrm{perm}(A)$ of $A$ as the expectation of a product of centered joint Gaussian random variables with a particular covariance matrix $C$. The algorithm outputs the empirical mean $S_{N}$ of this product after sampling $N$ times. Our algorithm runs in total time $O(M^{3} + M^{2}N + MN)$ with failure probability \begin{equation*} P(|S_{N}-\text{perm}(A)| > t) \leq \frac{3^{M}}{t^{2}N} \prod^{2M}_{i=1} C_{ii}. \end{equation*} In particular, we can estimate $\mathrm{perm}(A)$ to an additive error of $\epsilon\bigg(\sqrt{3^{2M}\prod^{2M}_{i=1} C_{ii}}\bigg)$ in polynomial time. We compare to a previous procedure due to Gurvits. We discuss how to find a particular $C$ using a semidefinite program and a relation to the Max-Cut problem and cut-norms.