Summary: 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 \(n^{1 - \epsilon}\) for any \(\epsilon>0\) unless \(\text{P}=\text{NP}\). 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 \(\Omega (f (n)\log^{2} n)\), or a directed cycle of length \(\Omega (f (n)\log n)\), for any nondecreasing, polynomial time computable function \(f\) in \(\omega (1)\). 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 \(\Omega (\log^{2} n /\log\log n)\) 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 \(\Omega (n)\) in directed expanders of constant bounded outdegree.
For the entire collection see [Zbl 1056.68007].