Bipartite Perfect Matching as a Real Polynomial
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We obtain a description of the Bipartite Perfect Matching decision problem as a multi-linear polynomial over the Reals. We show that it has full degree and has (1-o(1))· 2^n^2 monomials with non-zero coefficients. In contrast, we show that in the dual representation (switching the roles of 0 and 1) there are only 2^θ(n log n) monomials with non-zero coefficients. Our proof relies heavily on the fact that the lattice of graphs which are “matching-covered” is Eulerian.
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