Departmental Papers (CIS)

Document Type

Conference Paper


Postprint version. Published in Lecture Notes in Computer Science, Volume 3142, Automata, Languages and Programming, (ICALP 2004), pages 222-233.
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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.

Date of this Version

July 2004



Date Posted: 22 December 2005