Directed Isoperimetric Theorems for Boolean Functions on the Hypergrid and an $\widetilde{O}(n\sqrt{d})$ Monotonicity Tester

TL;DR

This paper introduces a new non-adaptive monotonicity tester for Boolean functions on hypergrids, achieving near-optimal query complexity by establishing generalized directed isoperimetric theorems.

Contribution

The authors develop the first non-adaptive, one-sided monotonicity tester for all constant n with query complexity near the lower bound, extending directed isoperimetric inequalities to hypergrids.

Findings

Achieved a $ ilde{O}(rac{n ext{d}}{ ext{varepsilon}^2})$ query complexity for monotonicity testing.

Generalized directed Talagrand inequalities from hypercube to hypergrid.

Provided new isoperimetric bounds that underpin the improved testing algorithm.

Abstract

The problem of testing monotonicity for Boolean functions on the hypergrid, $f:[n]^d \to \{0,1\}$ is a classic topic in property testing. When $n=2$, the domain is the hypercube. For the hypercube case, a breakthrough result of Khot-Minzer-Safra (FOCS 2015) gave a non-adaptive, one-sided tester making $\widetilde{O}(\varepsilon^{-2}\sqrt{d})$ queries. Up to polylog $d$ and $\varepsilon$ factors, this bound matches the $\widetilde{\Omega}(\sqrt{d})$-query non-adaptive lower bound (Chen-De-Servedio-Tan (STOC 2015), Chen-Waingarten-Xie (STOC 2017)). For any $n > 2$, the optimal non-adaptive complexity was unknown. A previous result of the authors achieves a $\widetilde{O}(d^{5/6})$-query upper bound (SODA 2020), quite far from the $\sqrt{d}$ bound for the hypercube. In this paper, we resolve the non-adaptive complexity of monotonicity testing for all constant $n$, up to $\text{poly}(\varepsilon^{-1}\log d)$ factors. Specifically, we give a non-adaptive, one-sided monotonicity tester making $\widetilde{O}(\varepsilon^{-2}n\sqrt{d})$ queries. From a technical standpoint, we prove new directed isoperimetric theorems over the hypergrid $[n]^d$. These results generalize the celebrated directed Talagrand inequalities that were only known for the hypercube.