Approximating Longest Directed Path

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 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 always finds a path of length Ω(log n), or a cycle of length Ω(log n), for any constant > 0 in these graphs. In contrast we show that there is a polynomial time algorithm always finding a path of length Ω(log n/ log log n) in these graphs. This separates the approximation hardness of Longest Path and Longest Cycle in this class of graphs. Furthermore, we present a polynomial time algorithm that finds paths of length Ω(n) in most digraphs of constant bounded outdegree.