The permanent of a non-negative square matrix can be well approximated by finding the minimum of the Bethe free energy functions associated with some suitably defined factor graph; the resulting approximation to the permanent is called the Bethe permanent. Vontobel gave a combinatorial characterization of the Bethe permanent via degree-$M$ Bethe permanents, which are based on degree-$M$ covers of the underlying factor graph. In this paper, we prove a degree-$M$-Bethe-permanent-based lower bound on the permanent of a non-negative matrix, which solves a conjecture proposed by Vontobel in [IEEE Trans. Inf. Theory, Mar. 2013]. We also prove a degree-$M$-Bethe-permanent-based upper bound on the permanent of a non-negative matrix. In the limit $M \to \infty$, these lower and upper bounds yield known Bethe-permanent-based lower and upper bounds on the permanent of a non-negative matrix. Moreover, we prove similar results for an approximation to the permanent known as the (scaled) Sinkhorn permanent.

翻译：非负方阵的永久值可以通过求解某个适当定义的因子图所对应的Bethe自由能函数的最小值来很好地逼近；由此得到的永久逼近称为Bethe永久。Vontobel通过基于底层因子图的$M$阶覆盖的$M$阶Bethe永久，给出了Bethe永久的一个组合刻画。本文中，我们证明了基于$M$阶Bethe永久的非负矩阵永久值的下界，这解决了Vontobel在[IEEE Trans. Inf. Theory, Mar. 2013]中提出的一个猜想。我们还证明了基于$M$阶Bethe永久的非负矩阵永久值的上界。当$M \to \infty$时，这些下界和上界退化为已知的基于Bethe永久的非负矩阵永久值的下界和上界。此外，我们针对另一种称为（缩放）Sinkhorn永久的永久逼近证明了类似的结果。