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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 n1-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) log2n), 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/ 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.
Björklund, Andreas; Husfeldt, Thore; and Khanna, Sanjeev, "Approximating Longest Directed Paths and Cycles" (2004). Departmental Papers (CIS). Paper 205.
Date Posted: 22 December 2005