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− for any > 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) log 2 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(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 Ω(n) in directed expanders of constant bounded outdegree.
With stronger assumptions the directed problem is hard to approximate to within a factor of n/ log 2+ n for any > 0 REF .
Approximating longest directed paths and cycles
In Proceedings of the 31st International Colloquium on Automata, Languages and Programming
Mathematics Computer Science