The Dual Polynomial of Bipartite Perfect Matching
Year: 2020
Type: preprint
Abstract: We obtain a description of the Boolean dual function of the Bipartite Perfect Matching decision problem, as a multilinear polynomial over the Reals. We show that in this polynomial, both the number of monomials and the magnitude of their coefficients are at most exponential in $\mathcal{O}(n \log n)$. As an application, we obtain a new upper bound of $\mathcal{O}(n^{1.5} \sqrt{\log n})$ on the approximate degree of the bipartite perfect matching function, improving the previous best known bound of $\mathcal{O}(n^{1.75})$. We deduce that, beyond a $\mathcal{O}(\sqrt{\log n})$ factor, the polynomial method cannot be used to improve the lower bound on the bounded-error quantum query complexity of bipartite perfect matching.
Source: arXiv (Cornell University)
Author: Gal Beniamini
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3023091520