Lower Bounds for Testing Directed Acyclicity in the Unidirectional Bounded-Degree Model
For researchers in property testing, this work provides tighter lower bounds for a fundamental graph property, advancing the understanding of query complexity in bounded-degree models.
The paper proves improved lower bounds for property testing of directed acyclicity in the unidirectional bounded-degree model: any one-sided ε-tester requires Ω̃(n^{2/3}) queries (improving from Ω̃(n^{5/9})), any two-sided ε-tester requires Ω(√n) queries (improving from Ω(n^{1/3})), and tolerant testing requires Ω(n) queries.
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.