Long Directed Detours: Reduction to $2$-Disjoint Paths

Abstract

In the Longest (s, t)-Detour problem, we look for an (s, t)-path that is at least k vertices longer than a shortest one. We study the parameterized complexity of Longest (s, t)-Detour when parameterized by k: this falls into the research paradigm of 'parameterization above guarantee'. Whereas the problem is known to be fixed-parameter tractable (FPT) on undirected graphs, the status of Longest (s, t)-Detour on directed graphs remains highly unclear: it is not even known to be solvable in polynomial time for k = 1. Recently, Fomin et al. made progress in this direction by showing that the problem is FPT on every class of directed graphs where the 3-Disjoint Paths problem is solvable in polynomial time. We improve upon their result by weakening this assumption: we show that only a polynomial-time algorithm for 2-Disjoint Paths is required. What is more, our approach yields an arguably simpler proof. * It has recently come to our attention that a similar result is given in [15]. Our work was conducted independently of this work.

Locally Rainbow Paths

“…Bezáková et al (2019) prove both variants to be fixed-parameter tractable, however, for the latter variant only on undirected graphs. While there has been some progress on directed graphs, most recently by Jacob, Włodarczyk, and Zehavi (2023), it is open whether finding a path with detour length at least one is polynomial-time solvable.…”