eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-07-07
54:1
54:14
10.4230/LIPIcs.ICALP.2017.54
article
Finding Detours is Fixed-Parameter Tractable
Bezáková, Ivona
Curticapean, Radu
Dell, Holger
Fomin, Fedor V.
We consider the following natural "above guarantee" parameterization of the classical longest path problem: For given vertices s and t of a graph G, and an integer k, the longest detour problem asks for an (s,t)-path in G that is at least k longer than a shortest (s,t)-path. Using insights into structural graph theory, we prove that the longest detour problem is fixed-parameter tractable (FPT) on undirected graphs and actually even admits a single-exponential algorithm, that is, one of running time exp(O(k)) * poly(n). This matches (up to the base of the exponential) the best algorithms for finding a path of length at least k.
Furthermore, we study a related problem, exact detour, that asks whether a graph G contains an (s,t)-path that is exactly k longer than a shortest (s,t)-path. For this problem, we obtain a randomized algorithm with running time about 2.746^k * poly(n), and a deterministic algorithm with running time about 6.745^k * poly(n), showing that this problem is FPT as well. Our algorithms for the exact detour problem apply to both undirected and directed graphs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol080-icalp2017/LIPIcs.ICALP.2017.54/LIPIcs.ICALP.2017.54.pdf
longest path
fixed-parameter tractable algorithms
above-guarantee parameterization
graph minors