# A Lower Bound on Cycle-Finding in Sparse Digraphs

## TL;DR

This paper establishes a new lower bound on the number of queries needed to find cycles in sparse directed graphs that are far from acyclic, improving previous bounds and advancing understanding of graph property testing.

## Contribution

It provides the first improved lower bound on cycle-finding query complexity in sparse digraphs, surpassing the classic a7a7a7a7a7a7a7a7a7a7a7a7a7a7a7a7a7a7a7a7a7a7a7a7a7 lower bound for this problem.

## Key findings

- Established an a7a9(N^{5/9}) query lower bound for cycle detection.
- Proved this bound applies to property testing of acyclicity in sparse digraphs.
- Improved upon the previous a7a9(\u221a N) lower bound.

## Abstract

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 $\tilde{\Omega}(N^{5/9})$ queries to an adjacency list representation of $G$. In the language of property testing, our result is an $\tilde{\Omega}(N^{5/9})$ 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 $\Omega(\sqrt{N})$ lower bound, implicit in Bender and Ron, which follows from a simple birthday paradox argument.