Simpler and faster algorithms for detours in planar digraphs

Abstract

In the DIRECTED DETOUR problem one is given a digraph G and a pair of vertices s and t, and the task is to decide whether there is a directed simple path from s to t in G whose length is larger than distG (s, t). The more general parameterized variant, DIRECTED LONG DETOUR, asks for a simple s-to-t path of length at least distG (s, t) + k, for a given parameter k. Surprisingly, it is still unknown whether DIRECTED DETOUR is polynomial-time solvable on general digraphs. However, for planar digraphs, Wu and Wang [Networks, '15] proposed an O(n3)-time algorithm for DIRECTED DETOUR, while Fomin et al. [STACS 2022] gave a 2O(κ) · nO(1)-time fpt algorithm for DIRECTED LONG DETOUR. The algorithm of Wu and Wang relies on a nontrivial analysis of how short detours may look like in a plane embedding, while the algorithm of Fomin et al. is based on a reduction to the 3-DISJOINT PATHS problem on planar digraphs. This latter problem is solvable in polynomial time using the algebraic machinery of Schrijver [SIAM J. Comp., '94], but the degree of the obtained polynomial factor is huge. In this paper we propose two simple algorithms: we show how to solve, in planar digraphs, DIRECTED DETOUR in time O(n2) and DIRECTED LONG DETOUR in time 2O(k) · n4 log n. In both cases, the idea is to reduce to the 2-DISJOINT PATHS problem in a planar digraph, and to observe that the obtained instances of this problem have a certain topological structure that makes them amenable to a direct greedy strategy. * This work is a part of project BOBR (KM, MP, MS) that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 948057). M. Hatzel was supported by the Federal Ministry of Education and Research (BMBF) and by a fellowship within the IFI programme of the German Academic Exchange Service (DAAD).