Boolean Unateness Testing with $\widetilde{O}(n^{3/4})$ Adaptive Queries

Abstract: We give an adaptive algorithm which tests whether an unknown Boolean function $f\colon {0, 1}^n \to{0, 1}$ is unate, i.e. every variable of $f$ is either non-decreasing or non-increasing, or $\epsilon$-far from unate with one-sided error using $\widetilde{O}(n^{3/4}/\epsilon^2)$ queries. This improves on the best adaptive $O(n/\epsilon)$-query algorithm from Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova and Seshadhri when $1/\epsilon \ll n^{1/4}$. Combined with the $\widetilde{\Omega}(n)$-query lower bound for non-adaptive algorithms with one-sided error of [CWX17, BCPRS17], we conclude that adaptivity helps for the testing of unateness with one-sided error. A crucial component of our algorithm is a new subroutine for finding bi-chromatic edges in the Boolean hypercube called adaptive edge search.