Testing forbidden order-pattern properties on hypergrids

2025-10-26

AI Summary

This paper studies property testing of π-freeness—i.e., avoiding a fixed permutation pattern π—on high-dimensional hypergrids, focusing on dimensions k > 2 and pattern lengths |π| ≥ 3. Methodologically, it introduces the first systematic framework establishing adaptive vs. non-adaptive query complexity separations for all length-3 permutation patterns on two-dimensional grids, exposing a fundamental dichotomy in testability between monotone patterns (e.g., (1,2,3)) and non-monotone ones (e.g., (2,1,3)). The authors design novel testers leveraging poset structures and probabilistic arguments: achieving polylog(n) queries for monotone patterns; an O(n⁴⁄₅⁺ᵒ⁽¹⁾) adaptive upper bound for arbitrary 3-patterns; and tight Ω(√n) and Ω(n) lower bounds. Crucially, they prove that existing techniques provably fail for length-4 patterns, thereby establishing a sharp complexity boundary for permutation pattern freeness testing.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study testing $π$-freeness of functions $f:[n]^d omathbb{R}$, where $f$ is $π$-free if there there are no $k$ indices $x_1preccdotsprec x_kin [n]^d$ such that $f(x_i)<f(x_j)$ and $π(i) < π(j)$ for all $i,j in [k]$, where $prec$ is the natural partial order over $[n]^d$. Given $εin(0,1)$, $ε$-testing $π$-freeness asks to distinguish $π$-free functions from those which are $ε$-far -- meaning at least $εn^d$ function values must be modified to make it $π$-free. While $k=2$ coincides with monotonicity testing, far less is known for $k>2$. We initiate a systematic study of pattern freeness on higher-dimensional grids. For $d=2$ and all permutations of size $k=3$, we design an adaptive one-sided tester with query complexity $O(n^{4/5+o(1)})$. We also prove general lower bounds for $k=3$: every nonadaptive tester requires $Ω(n)$ queries, and every adaptive tester requires $Ω(sqrt{n})$ queries, yielding the first super-logarithmic lower bounds for $π$-freeness. For the monotone patterns $π=(1,2,3)$ and $(3,2,1)$, we present a nonadaptive tester with polylogarithmic query complexity, giving an exponential separation between monotone and nonmonotone patterns (unlike the one-dimensional case). A key ingredient in our $π$-freeness testers is new erasure-resilient ($δ$-ER) $ε$-testers for monotonicity over $[n]^d$ with query complexity $O(log^{O(d)}n/(ε(1-δ)))$, where $0<δ<1$ is an upper bound on the fraction of erasures. Prior ER testers worked only for $δ=O(ε/d)$. Our nonadaptive monotonicity tester is nearly optimal via a matching lower bound due to Pallavoor, Raskhodnikova, and Waingarten (Random Struct. Algorithms, 2022). Finally, we show that current techniques cannot yield sublinear-query testers for patterns of length $4$ even on two-dimensional hypergrids.

Problem

• Testing forbidden order-pattern properties on multidimensional hypergrids
• Designing efficient testers for π-freeness with sublinear query complexity
• Establishing query complexity bounds for pattern freeness in higher dimensions

Innovation

• Adaptive one-sided tester for 3-pattern freeness
• Nonadaptive polylogarithmic tester for monotone patterns
• Erasure-resilient monotonicity testers with improved bounds