Finding Detours is Fixed-parameter Tractable

TL;DR

This paper proves that the Longest Detour problem, which asks for a path longer than the shortest by at least k, is fixed-parameter tractable with efficient algorithms for both undirected and directed graphs.

Contribution

It introduces fixed-parameter algorithms for Longest Detour and Exact Detour, providing the first such algorithms with exponential dependence on k.

Findings

Longest Detour is FPT on undirected graphs with exp(O(k)) time.

Exact Detour is FPT with randomized and deterministic algorithms.

Algorithms apply to both undirected and directed graphs.

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.