Beyond Talagrand Functions: New Lower Bounds for Testing Monotonicity and Unateness

TL;DR

This paper establishes new lower bounds on the query complexity for testing monotonicity and unateness of Boolean functions, surpassing previous bounds and matching recent upper bounds, using novel random function constructions.

Contribution

It introduces a new family of random Boolean functions to derive stronger lower bounds for testing monotonicity and unateness, improving upon prior results.

Findings

Lower bound of a(n^{1/3}) for monotonicity testing

Lower bound of a(n^{2/3}) for unateness testing

Lower bound of a(n) for non-adaptive unateness testing

Abstract

We prove a lower bound of $\tilde{\Omega}(n^{1/3})$ for the query complexity of any two-sided and adaptive algorithm that tests whether an unknown Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is monotone or far from monotone. This improves the recent bound of $\tilde{\Omega}(n^{1/4})$ for the same problem by Belovs and Blais [BB15]. Our result builds on a new family of random Boolean functions that can be viewed as a two-level extension of Talagrand's random DNFs. Beyond monotonicity, we also prove a lower bound of $\tilde{\Omega}(n^{2/3})$ for any two-sided and adaptive algorithm, and a lower bound of $\tilde{\Omega}(n)$ for any one-sided and non-adaptive algorithm for testing unateness, a natural generalization of monotonicity. The latter matches the recent linear upper bounds by Khot and Shinkar [KS15] and by Chakrabarty and Seshadhri [CS16].