Editing Open Problems:41

{{Header
|title=Testing Acyclicity
|source=bertinoro11
|who=Dana Ron
}}
Consider the problem of testing acyclicity in ''directed'' bounded-degree graphs
(in the incidence list model, where one can query both outgoing and
incoming edges).

'''Question:''' What is the best algorithm for this problem?

'''Background:''' There is a lower bound of $\Omega(n^{1/3})$
for adaptive, two-sided error algorithms, where $n$ is the number of vertices {{cite|BenderR-02}}. 
No sublinear upper bound is known. (For dense graphs, in the adjacency matrix model, one can test
the property using $\operatorname{poly}(1/\epsilon)$ queries.) The best known lower bound for 1-sided error testing is only $\Omega(\sqrt{n})$.