In a recent paper, Beniamini and Nisan gave a closed-form formula for the unique multilinear polynomial for the Boolean function determining whether a given bipartite graph $G \subseteq K_{n,n}$ has a perfect matching, together with an efficient algorithm for computing the coefficients of the monomials of this polynomial. We give the following generalization: Given an arbitrary non-negative weight function $w$ on the edges of $K_{n,n}$, consider its set of minimum weight perfect matchings. We give the real multilinear polynomial for the Boolean function which determines if a graph $G \subseteq K_{n,n}$ contains one of these minimum weight perfect matchings.

中文翻译：

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二部图最小权重完美匹配的实数多项式

在最近的一篇论文中，Beniamini 和 Nisan 给出了布尔函数的唯一多重线性多项式的闭式公式，以确定给定的二部图 $G \subseteq K_{n,n}$ 是否具有完美匹配，以及一个有效的算法用于计算该多项式的单项式的系数。我们给出以下概括： 给定 $K_{n,n}$ 边上的任意非负权重函数 $w$，考虑其最小权重完美匹配集。我们给出布尔函数的实多线性多项式，该函数确定图 $G \subseteq K_{n,n}$ 是否包含这些最小权重完美匹配之一。