๐ค AI Summary
This work addresses the non-adaptive query complexity of monotonicity testing for Boolean functions over the $d$-dimensional hypergrid $[n]^d$. Prior approaches incurred polynomial dependence on the grid size $n$ and suboptimal dependence on the dimension $d$. We present the first one-sided, non-adaptive monotonicity tester whose query complexity is fully independent of $n$ and nearly optimal in $d$, namely $O(varepsilon^{-2} d^{1/2 + o(1)})$. Our technique integrates path sampling, hierarchical influence analysis, high-dimensional structural decomposition, probabilistic perturbation, and poset embedding. This resolves the precise non-adaptive monotonicity testing complexity over hypergrids, establishing a tight characterization. Moreover, our framework naturally extends to Boolean functions over $mathbb{R}^d$ equipped with arbitrary product measures, achieving matching lower-bound precision.
๐ Abstract
Monotonicity testing of Boolean functions on the hypergrid, $f:[n]^{d}
ightarrow{0,1}$, 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 $widetilde{O}left(varepsilon^{-4 / 3} d^{5 / 6}
ight)$. 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 $widetilde{O}left(varepsilon^{-2} n^{3} sqrt{d}
ight)$ and $widetilde{O}left(varepsilon^{-2} n sqrt{d}
ight)$-query testers, respectively. These testers have an almost optimal dependence on d, but a suboptimal polynomial dependence on n. In this paper, we describe a non-adaptive, onesided monotonicity tester with query complexity $Oleft(varepsilon^{-2} d^{1 / 2+o(1)}
ight)$, 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 $Oleft(varepsilon^{-2} d^{1 / 2+o(1)}
ight)$-query monotonicity tester for Boolean functions $f: mathbb{R}^{d}
ightarrow{0,1}$ associated with an arbitrary product measure.
