Approximating Longest Directed Paths and Cycles

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. 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 This conference paper is available at ScholarlyCommons: http://repository.upenn.edu/cis_papers/205 Approximating Longest Directed Paths and Cycles Andreas Björklund, Thore Husfeldt, and Sanjeev Khanna 1 Department of Computer Science, Lund University, Box 118, 221 00 Lund, Sweden. thore@cs.lu.se 2 Dept. of CIS, University of Pennsylvania, Philadelphia, PA 19104. sanjeev@cis.upenn.edu 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− for any > 0 unless P = NP. In particular, the result holds for digraphs of constant bounded outdegree that contain a Hamiltonian cycle. 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− for any > 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) log n), 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(log 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.