A Lower Bound on Cycle-Finding in Sparse Digraphs

We consider the problem of finding a cycle in a sparse directed graph G that is promised to be far from acyclic, meaning that the smallest feedback arc set in G is large. We prove an information-theoretic lower bound, showing that for N -vertex graphs with constant outdegree any algorithm for this problem must make Ω(N 5/9 ) queries to an adjacency list representation of G. In the language of property testing, our result is an Ω(N 5/9 ) lower bound on the query complexity of one-sided algorithms for testing whether sparse digraphs with constant outdegree are far from acyclic. This is the first improvement on the Ω( √ N ) lower bound, implicit in Bender and Ron [BR02], which follows from a simple birthday paradox argument.