Boolean function monotonicity testing requires (almost) $n^{1/2}$ non-adaptive queries

TL;DR

This paper establishes a near-optimal lower bound on the number of non-adaptive queries needed to test Boolean function monotonicity, significantly advancing understanding of query complexity in property testing.

Contribution

It proves a nearly tight lower bound of (n^{1/2 - c}) for non-adaptive monotonicity testing, improving previous bounds and conjecturing optimality.

Findings

Lower bound of (n^{1/2 - c}) for all c>0.

Improves previous (n^{1/5}) lower bound.

Suggests (n^{1/2}) as the optimal bound.

Abstract

We prove a lower bound of $\Omega(n^{1/2 - c})$, for all $c>0$, on the query complexity of (two-sided error) non-adaptive algorithms for testing whether an $n$-variable Boolean function is monotone versus constant-far from monotone. This improves a $\tilde{\Omega}(n^{1/5})$ lower bound for the same problem that was recently given in [CST14] and is very close to $\Omega(n^{1/2})$, which we conjecture is the optimal lower bound for this model.