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arXiv · 1607.07737

Finding Detours is Fixed-parameter Tractable

Ivona Bezáková·Radu Curticapean·Holger Dell·Fedor V. Fomin
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Abstract

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 problem Longest Detour 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 Longest Detour 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 the 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, and a deterministic algorithm with running time about 6.745^k, showing that this problem is FPT as well. Our algorithms for Exact Detour apply to both undirected and directed graphs.

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Ivona Bezáková, Radu Curticapean, Holger Dell, Fedor V. Fomin. 2016-07-26. Finding Detours is Fixed-parameter Tractable. https://doi.org/10.4230/lipics.icalp.2017.54

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