Thore Husfeldt
News
New Paper with Björklund and Khanna
“Approximating longest directed paths and cycles,” to be presented at ICALP 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)log2 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(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.
Extended abstract under On-line papers.
Thu, 29 Apr 2004 | Category: News | Permanent link