A $d^{1/2+o(1)}$ Monotonicity Tester for Boolean Functions on $d$-Dimensional Hypergrids

TL;DR

The paper resolves the non-adaptive, one-sided monotonicity testing problem for Boolean functions on $[n]^d$ by constructing a non-adaptive tester with query complexity $O(\varepsilon^{-2} d^{1/2+o(1)})$ that is independent of $n$, extending to product-measure spaces on $\mathbb{R}^d$. The core technique augments directed random-walk testers with coordinated shift strategies, enabling robust analysis on hypergrids via a red/blue subgraph framework, translations, and persistence-based peeling arguments. A key technical advance is handling hypergrid geometry through seed-regular violation subgraphs and the persist-or-blow-up lemma, ensuring down/up-persistence and enabling the main rejection guarantees. The results bridge prior hypercube and hypergrid monotonicity-testing bounds, offering near-optimal $d$-dependence under non-adaptivity and laying groundwork for further refinements toward a $\tilde{O}(\varepsilon^{-2}\sqrt{d})$ bound.

Abstract

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 $n$. In this paper, we describe a non-adaptive, one-sided monotonicity tester with query complexity $O(\varepsilon^{-2} d^{1/2 + o(1)})$, 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.

Figures (1)

• Figure 1: This figure shows the key argument that either up-walks + downshifts, or down-walks find violations. The edge $(\mathbf{x},\mathbf{y})$ is in the initial violation matching. Parallel curves of the same shape denote the same shift. So $\mathbf{x}' = \mathbf{x}+\mathbf{s}$, $\mathbf{y}' = \mathbf{y} + \mathbf{s}$, and $\mathbf{z}' = \mathbf{z} + \mathbf{s}$. Similarly, we see both $\mathbf{x}$ and $\mathbf{z}'$ shifted below by $\mathbf{t}$. The $1$-valued points are colored black and the $0$-valued points are colored white. Gray points do not have an a priori guarantee on function value. If $\mathbf{z}'$ is $\mathsf{mzb}$, then $f(\mathbf{z}'-\mathbf{t}) = 0$ with high probability. In this case, $(\mathbf{x}-\mathbf{t}, \mathbf{z}'-\mathbf{t})$ is a likely violation. If not, then $(\mathbf{z}'-\mathbf{t},\mathbf{y}')$ is a likely violation.

Theorems & Definitions (103)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3: Hypergrid Walk Distribution
• Definition 1.4: Shift Distributions
• Remark 1.5
• Theorem 1.6: Main Theorem
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4: Theorem 1.4, BlChSe23