A Real Polynomial for Bipartite Graph Minimum Weight Perfect Matchings
Abstract
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 ⊂eq 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 non-negative 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 ⊂eq Kn,n contains one of these minimum weight perfect matchings.
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