Algebraic Representations of Unique Bipartite Perfect Matching

Abstract: We obtain complete characterizations of the Unique Bipartite Perfect Matching function, and of its Boolean dual, using multilinear polynomials over the reals. Building on previous results, we show that, surprisingly, the dual description is sparse and has low $\ell_1$-norm -- only exponential in $\Theta(n \log n)$, and this result extends even to other families of matching-related functions. Our approach relies on the M\"obius numbers in the matching-covered lattice, and a key ingredient in our proof is M\"obius' inversion formula. These polynomial representations yield complexity-theoretic results. For instance, we show that unique bipartite matching is evasive for classical decision trees, and nearly evasive even for generalized query models. We also obtain a tight $\Theta(n \log n)$ bound on the log-rank of the associated two-party communication task.