A real polynomial for bipartite graph minimum weight perfect matchings

In a recent paper, Beniamini and Nisan [4] gave a closed-form formula for the unique multilinear polynomial for the Boolean function determining whether a given bipartite graph G subset of Kn,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 weight function w on the edges of Kn,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 subset of Kn,n contains one of these minimum weight perfect matchings. Finally, we discuss a number of open problems which follow from [4] and our work; in particular, extending the main theorem of [4] to non-bipartite graphs.(c) 2022 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).