Lower Bounds for Testing Directed Acyclicity in the Unidirectional Bounded-Degree Model

2026-04-15

AI Summary

This work investigates the query complexity limits of efficiently testing acyclicity in directed graphs with bounded maximum out-degree. It improves existing lower bounds on query complexity from Õ(n^{5/9}) and Ω(n^{1/3}) to Õ(n^{2/3}) and Ω(√n) for one-sided and two-sided error models, respectively, and establishes the first Ω(n) lower bound for tolerant acyclicity testing. The results are achieved through a combination of reduction techniques—such as from bounded-degree 3-colorability—and combinatorial constructions. These contributions substantially tighten the theoretical bounds for acyclicity testing in directed graphs and deepen the understanding of local testing complexity in this setting.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Applications of cryptography
• 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 property testing of directed acyclicity in the unidirectional bounded-degree oracle model, where a query to a vertex reveals its outgoing neighbors. We prove that there exist absolute constants $d_0\in\mathbb{N}$ and $\varepsilon>0$ such that for every constant $d\ge d_0$, any one-sided $\varepsilon$-tester for acyclicity on $n$-vertex digraphs of maximum outdegree at most $d$ requires $\widetildeΩ(n^{2/3})$ queries. This improves the previous $\widetildeΩ(n^{5/9})$ lower bound for one-sided testing of acyclicity in the same model. We also prove that, under the same degree assumption, any two-sided $\varepsilon$-tester requires $Ω(\sqrt n)$ queries, improving the previous $Ω(n^{1/3})$ lower bound. We further prove an $Ω(n)$ lower bound for tolerant testing for some absolute constant outdegree bound $d$ by reduction from bounded-degree $3$-colorability.

Problem

• directed acyclicity
• property testing
• bounded-degree model
• query complexity
• lower bounds

Innovation

• property testing
• directed acyclicity
• query complexity
• lower bounds
• bounded-degree graphs