Bipartite Perfect Matching as a Real Polynomial

📝 Original Info

• Title: Bipartite Perfect Matching as a Real Polynomial
• ArXiv ID: 2001.07642
• Date: 2020-02-25
• Authors: Gal Beniamini and Noam Nisan

📝 Abstract

We obtain a description of the Bipartite Perfect Matching decision problem as a multilinear polynomial over the Reals. We show that it has full degree and $(1-o_n(1))\cdot 2^{n^2}$ monomials with non-zero coefficients. In contrast, we show that in the dual representation (switching the roles of 0 and 1) the number of monomials is only exponential in $\Theta(n \log n)$. Our proof relies heavily on the fact that the lattice of graphs which are "matching-covered" is Eulerian.

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