We consider the permanent of a square matrix with non-negative entries. A tractable approximation is given by the so-called Bethe permanent that can be efficiently computed by running the sum-product algorithm on a suitable factor graph. While the ratio of the permanent of a matrix to its Bethe permanent is, in the worst case, upper and lower bounded by expressions that are exponentially far apart in the matrix size, in practice it is observed for many ensembles of matrices of interest that this ratio is strongly concentrated around some value that depends only on the matrix size. In this paper, for an ensemble of block-structured matrices where entries in a block take the same value, we numerically study the ratio of the permanent of a matrix to its Bethe permanent. It is observed that also for this ensemble the ratio is strongly concentrated around some value depending only on a few key parameters of the ensemble. We use graph-cover-based approaches to explain the reasons for this behavior and to quantify the observed value.

翻译：我们考虑具有非负元素的方阵的永久性。一种易于处理的近似由所谓的Bethe永久性给出，该值可通过在合适的因子图上运行和积算法高效计算。虽然矩阵永久性与其Bethe永久性之比在最坏情况下被矩阵尺寸指数级分离的上下界所限定，但在实际应用中观察到，对于许多重要矩阵集合，该比值强烈集中在仅取决于矩阵尺寸的某个值附近。本文针对块结构矩阵集合（其中块内元素取值相同），数值研究了矩阵永久性与其Bethe永久性之比。观测发现该集合的比值同样强烈集中在仅取决于集合少数关键参数的某个值附近。我们采用基于图覆盖的方法来解释此行为的原因，并对观测值进行量化分析。