Approximating Longest Directed Paths and Cycles

Abstract

We investigate the hardness of approximating the longest path and the longest cycle in directed graphs on n vertices. We show that neither of these two problems can be polynomial time approximated within n ^{1 − − ε} for any ε> 0 unless P=NP. In particular, the result holds for digraphs of constant bounded outdegree that contain a Hamiltonian cycle.

Assuming the stronger complexity conjecture that Satisfiability cannot be solved in subexponential time, we show that there is no polynomial time algorithm that finds a directed path of length Ω(f(n)log² n), or a directed cycle of length Ω(f(n)log n), for any nondecreasing, polynomial time computable function f in ω(1). With a recent algorithm for undirected graphs by Gabow, this shows that long paths and cycles are harder to find in directed graphs than in undirected graphs.

We also find a directed path of length Ω(log² n/loglog n) in Hamiltonian digraphs with bounded outdegree. With our hardness results, this shows that long directed cycles are harder to find than a long directed paths. Furthermore, we present a simple polynomial time algorithm that finds paths of length Ω(n) in directed expanders of constant bounded outdegree.