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, i.e., a subset of edges whose deletion results in an acyclic graph, 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 Ω̄(N5/9) queries to an adjacency list representation of G. In the language of property testing, our result is an Ω̄(N5/9) 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 the work of Bender and Ron, which follows from a simple birthday paradox argument.