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Approximating longest directed paths and cycles

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Author

Summary, in English

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-epsilon)for any epsilon > 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 Omega(f(n) log(2) n), or a directed cycle of length Omega(f(n) log n), for any nondecreasing, polynomial time computable function f in w(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 Omega(log(2) 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 Omega(n) in directed expanders of constant bounded outdegree.

Publishing year

2004

Language

English

Pages

222-233

Publication/Series

Lecture Notes in Computer Science

Volume

3142

Document type

Journal article

Publisher

Springer

Topic

  • Computer Science

Status

Published

ISBN/ISSN/Other

  • ISSN: 1611-3349