A o(n) monotonicity tester for Boolean functions over the hypercube

TL;DR

The paper introduces an efficient randomized, non-adaptive monotonicity tester for Boolean functions over the hypercube, requiring significantly fewer queries than previous methods, with an optimal linearithmic dependence on the dimension.

Contribution

It presents the first monotonicity tester with query complexity $O(n^{7/8} ext{poly}(rac{1}{ ext{eps}}) ext{log}(1/ ext{eps}))$, improving over prior approaches.

Findings

Query complexity is sublinear in the dimension n.

Tester is non-adaptive and one-sided, simplifying implementation.

Achieves near-optimal dependence on the dimension n.

Abstract

A Boolean function $f:\{0,1\}^n \mapsto \{0,1\}$ is said to be $\eps$-far from monotone if $f$ needs to be modified in at least $\eps$-fraction of the points to make it monotone. We design a randomized tester that is given oracle access to $f$ and an input parameter $\eps>0$, and has the following guarantee: It outputs {\sf Yes} if the function is monotonically non-decreasing, and outputs {\sf No} with probability $>2/3$, if the function is $\eps$-far from monotone. This non-adaptive, one-sided tester makes $O(n^{7/8}\eps^{-3/2}\ln(1/\eps))$ queries to the oracle.