Algebraic Representations of Unique Bipartite Perfect Matching

TL;DR

This paper provides algebraic characterizations of the Unique Bipartite Perfect Matching function and its dual, revealing their polynomial representations and implications for computational complexity.

Contribution

It introduces complete polynomial characterizations of the matching function and its dual, with bounds on sparsity and norm, extending to related functions and complexity results.

Findings

Dual description is sparse with low $\

Unique bipartite matching is evasive for decision trees.

Log-rank of the associated communication problem is tightly bounded by $\

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.