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

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 $\widetilde{\Omega}(n^{2/3})$ queries. This improves the previous $\widetilde{\Omega}(n^{5/9})$ 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 $\Omega(\sqrt n)$ queries, improving the previous $\Omega(n^{1/3})$ lower bound. We further prove an $\Omega(n)$ lower bound for tolerant testing for some absolute constant outdegree bound $d$ by reduction from bounded-degree $3$-colorability.