A Real Polynomial for Bipartite Graph Minimum Weight Perfect Matchings

03/19/2020

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by   Karthik Gajulapalli, et al.

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In a recent paper, Beniamini and Nisan <cit.> gave a closed-form formula for the unique multilinear polynomial for the Boolean function determining whether a given bipartite graph G ⊆ K_n,n has a perfect matching, together with an efficient algorithm for computing its terms. 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 ⊆ K_n,n contains one of these minimum weight perfect matchings.