Simpler and faster algorithms for detours in planar digraphs

TL;DR

This paper introduces simpler, faster algorithms for the directed detour and long detour problems in planar digraphs, improving computational efficiency by leveraging topological structures and reductions to disjoint paths problems.

Contribution

It presents two new algorithms for planar digraphs: an $ ext{O}(n^2)$-time solution for directed detour and an $ ext{O}(2^{O(k)} imes n^4 imes ext{log} n)$-time solution for directed long detour, simplifying previous approaches.

Findings

Directed detour solved in $ ext{O}(n^2)$ time.

Directed long detour solved in $2^{O(k)} imes n^4 imes ext{log} n$ time.

Reduction to 2-disjoint paths with exploitable topological structure.

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 $\mathsf{dist}_{G}(s,t)$. The more general parameterized variant, directed long detour, asks for a simple $s$-to-$t$ path of length at least $\mathsf{dist}_{G}(s,t)+k$, 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 $\mathcal{O}(n^3)$-time algorithm for directed detour, while Fomin et al.~[STACS 2022] gave a $2^{\mathcal{O}(k)}\cdot n^{\mathcal{O}(1)}$-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 ${\S}$-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 $\mathcal{O}(n^2)$ and directed long detour in time $2^{\mathcal{O}(k)}\cdot n^4 \log n$. 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.