Near-Linear Lower Bounds for Testing Directed Acyclicity

We study testing acyclicity of directed graphs in the bounded-degree incidence-lists model, where a tester can query the outneighbors of a vertex but not its inneighbors. Despite significant effort, no tester with $o(n)$ queries is known, while the best established lower bounds are $\widetildeΩ(n^{5/9})$ for one-sided testers (Chen, Randolph, Servedio, and Sun, 2022) and $Ω(n^{1/3})$ for two-sided testers (Bender and Ron, 2002). We show that a query exponent arbitrarily close to one is necessary: for every fixed $η\in(0,1)$, there exists a constant $\varepsilon_η>0$ such that, for every constant $d\ge2$, every randomized adaptive two-sided $\varepsilon_η$-tester for acyclicity of simple $n$-vertex directed graphs whose indegrees and outdegrees are at most $d$ makes $Ω(n^{1-η})$ queries. Our proof overcomes two barriers. First, in previous constructions, the acyclic part of the graph is easy to recognize, which helps an algorithm stay inside the cyclic part. We hide a high-girth cyclic core inside a hierarchy of acyclic masking regions, in which each region imitates the levels below it and can be told apart from them only through rare events, and an iterated information-theoretic argument shows that every exploration observes only small, low-diameter forest components of the core. Second, a two-sided tester may exploit the statistics of collisions without ever seeing a cycle, and at a near-linear budget collisions are unavoidable. We compare complete transcripts with those of an acyclic input built on a fixed core by counting injective embeddings of the observed forest: in high-girth cores, all connected collision patterns of bounded order contribute identically, which makes the two transcript distributions statistically indistinguishable.