ScholarlyCommons

Title

Approximating Longest Directed Paths and Cycles

Author(s)

Andreas Björklund, Lund University
Thore Husfeldt, Lund University
Sanjeev Khanna, University of PennsylvaniaFollow

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.

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

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-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.