Approximating the Distance to Monotonicity of Boolean Functions

TL;DR

This paper presents a nonadaptive algorithm that estimates how far a Boolean function is from monotonicity with a poly(n, 1/alpha) query complexity, and establishes tight lower bounds showing the limits of such algorithms.

Contribution

It introduces a poly(n, 1/alpha) query nonadaptive algorithm for approximating the distance to monotonicity and proves matching lower bounds for nonadaptive algorithms, resolving an open question.

Findings

The algorithm achieves an (\u221A n) approximation.

Any nonadaptive algorithm with better approximation requires exponentially many queries.

The lower bounds extend to unateness and k-junta properties.

Abstract

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.