We design a nonadaptive algorithm that, given oracle access to a function $f: \{ 0,1\}^n \to \{ 0,1\}$ which is $\alpha$-far from monotone, makes poly$(n, 1/\alpha)$ queries and returns an estimate that, with high probability, is an $\widetilde{ O}(\sqrt{ n})$-approximation to the distance of $f$ to monotonicity. The analysis of our algorithm relies on an improvement to the directed isoperimetric inequality of Khot, Minzer, and Safra (SIAM J. Comput., 2018). Furthermore, we rule out a poly$(n, 1/\alpha)$-query nonadaptive algorithm that approximates the distance to monotonicity significantly better by showing that, for all constant $\kappa > 0,$ every nonadaptive $n^{ 1/2 - \kappa}$-approximation algorithm for this problem requires $2^{ n^\kappa}$ queries. This answers a question of Seshadhri (Property Testing Review, 2014) for the case of nonadaptive algorithms. We obtain our lower bound by proving an analogous bound for erasure-resilient (and tolerant) testers. Our method also yields the same lower bounds for unateness and being a $k$-junta.
Categories: cs.DS cs.CC cs.DM