A $o(d) \cdot \text{polylog}~n$ Monotonicity Tester for Boolean Functions over the Hypergrid $[n]^d$

TL;DR

This paper introduces a new monotonicity tester for Boolean functions over hypergrids with query complexity nearly linear in dimension, improving previous methods especially for certain ranges of n.

Contribution

The paper presents a non-adaptive, one-sided error monotonicity tester with sublinear query complexity for hypergrid domains, utilizing an augmented hypergrid structure and a novel isoperimetric inequality.

Findings

Achieves query complexity of ^{5/6} polylog(n,1/)

Works effectively when n = 2^{d^{o(1)}}

Introduces a Margulis-style isoperimetric result for augmented hypergrids

Abstract

We study monotonicity testing of Boolean functions over the hypergrid $[n]^d$ and design a non-adaptive tester with $1$-sided error whose query complexity is $\tilde{O}(d^{5/6})\cdot \text{poly}(\log n,1/\epsilon)$. Previous to our work, the best known testers had query complexity linear in $d$ but independent of $n$. We improve upon these testers as long as $n = 2^{d^{o(1)}}$. To obtain our results, we work with what we call the augmented hypergrid, which adds extra edges to the hypergrid. Our main technical contribution is a Margulis-style isoperimetric result for the augmented hypergrid, and our tester, like previous testers for the hypercube domain, performs directed random walks on this structure.