
Departmental Papers (CIS)
Date of this Version
July 2004
Document Type
Conference Paper
Recommended Citation
Andreas Björklund, Thore Husfeldt, and Sanjeev Khanna, "Approximating Longest Directed Paths and Cycles", . July 2004.
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 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 Posted: 22 December 2005
Comments
Postprint version. Published in Lecture Notes in Computer Science, Volume 3142, Automata, Languages and Programming, (ICALP 2004), pages 222-233.
Publisher URL: http://dx.doi.org/10.1007/b99859