The Approximate Degree of Bipartite Perfect Matching

The approximate degree of a Boolean function is the least degree of a Real multilinear polynomial approximating it in the $\ell_\infty$-norm over the Boolean hypercube. We consider the Bipartite Perfect Matching function, which is the indicator over all bipartite graphs having a perfect matching, and show that its approximate degree is $\widetilde{\Theta}(n^{1.5})$. The upper bound is obtained by fully characterizing the unique multilinear polynomial representing the Boolean dual of the perfect matching function, over the Reals. In particular, we show that this polynomial has very small $\ell_1$-norm -- only exponential in $\Theta(n \log n)$. The lower bound follows by bounding the spectral sensitivity of the perfect matching function, which is the spectral radius of its cut-graph on the hypercube \cite{aaronson2020degree, huang2019induced}. We show that the spectral sensitivity of perfect matching is exactly $\Theta(n^{1.5})$.