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.

翻译：我们提出一种随机算法，用于估计 $M \times M$ 实矩阵 $A$ 的积和式，直至加性误差。通过将 $A$ 的积和式 $\mathrm{perm}(A)$ 视作具有特定协方差矩阵 $C$ 的中心联合高斯随机变量乘积的期望值来实现。该算法在采样 $N$ 次后输出该乘积的经验均值 $S_{N}$。算法总运行时间为 $O(M^{3} + M^{2}N + MN)$，失败概率满足 \begin{equation*} P(|S_{N}-\text{perm}(A)| > t) \leq \frac{3^{M}}{t^{2}N} \prod^{2M}_{i=1} C_{ii}. \end{equation*} 特别地，我们可在多项式时间内以 $\epsilon\bigg(\sqrt{3^{2M}\prod^{2M}_{i=1} C_{ii}}\bigg)$ 的加性误差估计 $\mathrm{perm}(A)$。我们与 Gurvits 提出的先前方法进行了比较，并讨论了如何通过半定规划确定特定 $C$ 及其与 Max-Cut 问题和割范数的关联。