Search for: All records

Monotonicity testing of Boolean functions on the hypergrid, $$f:[n]^d \to \{0,1\}$$, is a classic topic in property testing. Determining the non-adaptive complexity of this problem is an important open question. For arbitrary $$n$$, [Black-Chakrabarty-Seshadhri, SODA 2020] describe a tester with query complexity $$\widetilde{O}(\varepsilon^{-4/3}d^{5/6})$$. This complexity is independent of $$n$$, but has a suboptimal dependence on $$d$$. Recently, [Braverman-Khot-Kindler-Minzer, ITCS 2023] and [Black-Chakrabarty-Seshadhri, STOC 2023] describe $$\widetilde{O}(\varepsilon^{-2} n^3\sqrt{d})$$ and $$\widetilde{O}(\varepsilon^{-2} n\sqrt{d})$$-query testers, respectively. These testers have an almost optimal dependence on $$d$$, but a suboptimal polynomial dependence on $$n$$. \smallskip In this paper, we describe a non-adaptive, one-sided monotonicity tester with query complexity $$O(\varepsilon^{-2} d^{1/2 + o(1)})$$, \emph{independent} of $$n$$. Up to the $$d^{o(1)}$$-factors, our result resolves the non-adaptive complexity of monotonicity testing for Boolean functions on hypergrids. The independence of $$n$$ yields a non-adaptive, one-sided $$O(\varepsilon^{-2} d^{1/2 + o(1)})$$-query monotonicity tester for Boolean functions $$f:\mathbb{R}^d \to \{0,1\}$$ associated with an arbitrary product measure. 

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 $$\otilde(\eps^{-2}\sqrt{d})$$ queries. Up to polylog $$d$$ and $$\eps$$ 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 $$\otilde(d^{5/6})$$-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 $$\poly(\eps^{-1}\log d)$$ factors. Specifically, we give a non-adaptive, one-sided monotonicity tester making $$\otilde(\eps^{-2}n\sqrt{d})$$ 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.