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

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 $\widetilde{O}(\varepsilon<sup>{-2}\sqrt{d})$ 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 $n &gt; 2$, the optimal non-adaptive complexity was unknown. A previous result of the authors achieves a $\widetilde{O}(d<sup>{5/6})$-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 $\widetilde{O}(\varepsilon<sup>{-2}n\sqrt{d})$ queries. From a technical standpoint, we prove new directed isoperimetric theorems over the hypergrid $[n]<sup>d$. These results generalize the celebrated directed Talagrand inequalities that were only known for the hypercube.