Testing Monotonicity of Real-Valued Functions on DAGs

2026-02-16

AI Summary

This study addresses the problem of monotonicity testing for real-valued functions over directed acyclic graphs (DAGs), where the goal is to determine whether a function is non-decreasing along every edge. By constructing a novel family of positive-match Ruzsa–Szemerédi graphs and combining transpose reductions with an analysis of closure edge counts, the authors establish the first Ω(√n) query lower bound for randomized adaptive one-sided testers on explicit bipartite DAGs. Furthermore, for any constant δ > 0, they prove an Ω(n^{1/2−δ}/√ε) lower bound for non-adaptive two-sided testers. Complementing these hardness results, they design an efficient non-adaptive one-sided algorithm tailored for sparse graphs, which outperforms the classical O(√(n/ε)) approach when mℓ = o(n³) or m = o(n^{3/2}), where m and ℓ denote the numbers of edges and maximal paths, respectively.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We study monotonicity testing of real-valued functions on directed acyclic graphs (DAGs) with $n$ vertices. For every constant $δ>0$, we prove a $Ω(n^{1/2-δ}/\sqrt{\varepsilon})$ lower bound against non-adaptive two-sided testers on DAGs, nearly matching the classical $O(\sqrt{n/\varepsilon})$-query upper bound. For constant $\varepsilon$, we also prove an $Ω(\sqrt n)$ lower bound for randomized adaptive one-sided testers on explicit bipartite DAGs, whereas previously only an $Ω(\log n)$ lower bound was known. A key technical ingredient in both lower bounds is positive-matching Ruzsa--Szemerédi families. On the algorithmic side, we give simple non-adaptive one-sided testers with query complexity $O(\sqrt{m\,\ell}/(\varepsilon n))$ and $O(m^{1/3}/\varepsilon^{2/3})$, where $m$ is the number of edges in the transitive reduction and $\ell$ is the number of edges in the transitive closure. For constant $\varepsilon>0$, these improve over the previous $O(\sqrt{n/\varepsilon})$ bound when $m\ell=o(n^3)$ and $m=o(n^{3/2})$, respectively.

Problem

• monotonicity testing
• real-valued functions
• directed acyclic graphs
• property testing

Innovation

• monotonicity testing
• directed acyclic graphs
• query complexity
• Ruzsa–Szemerédi families
• non-adaptive testers