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Detours in Directed Graphs

Authors Fedor V. Fomin , Petr A. Golovach , William Lochet , Danil Sagunov , Kirill Simonov, Saket Saurabh

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Author Details

Fedor V. Fomin
  • Department of Informatics, University of Bergen, Norway
Petr A. Golovach
  • Department of Informatics, University of Bergen, Norway
William Lochet
  • Department of Informatics, University of Bergen, Norway
Danil Sagunov
  • St. Petersburg Department of V.A. Steklov Institute of Mathematics, Russia
  • JetBrains Research, Saint Petersburg, Russia
Kirill Simonov
  • Algorithms and Complexity Group, TU Wien, Austria
Saket Saurabh
  • Institute of Mathematical Sciences, HBNI, Chennai, India
  • Department of Informatics, University of Bergen, Norway

Funding

  • Fomin, Fedor V.: supported by the Research Council of Norway via the project BWCA (grant no. 314528).
  • Golovach, Petr A.: supported by the Research Council of Norway via the project BWCA (grant no. 314528).
  • Lochet, William: supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant no. 819416).
  • Sagunov, Danil: supported by Leonhard Euler International Mathematical Institute in Saint Petersburg (agreement no. 075-15-2019-1620).
  • Simonov, Kirill: supported by the Austrian Science Fund (FWF) via project Y1329 (Parameterized Analysis in Artificial Intelligence).
  • Saurabh, Saket: supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant no. 819416) and Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18.

Cite As Get BibTex

Fedor V. Fomin, Petr A. Golovach, William Lochet, Danil Sagunov, Kirill Simonov, and Saket Saurabh. Detours in Directed Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.STACS.2022.29

Abstract

We study two "above guarantee" versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, the task is to decide whether a graph has an (s,t)-path of length at least dist_G(s,t)+k (where dist_G(s,t) denotes the length of a shortest path from s to t). Bezáková et al. [Ivona Bezáková et al., 2019] proved that on undirected graphs the problem is fixed-parameter tractable (FPT) by providing an algorithm of running time 2^{O(k)}⋅ n. Further, they left the parameterized complexity of the problem on directed graphs open. Our first main result establishes a connection between Longest Detour on directed graphs and 3-Disjoint Paths on directed graphs. Using these new insights, we design a 2^{O (k)}· n^{O(1)} time algorithm for the problem on directed planar graphs. Further, the new approach yields a significantly faster FPT algorithm on undirected graphs. 
In the second variant of Longest Path, namely Longest Path above Diameter, the task is to decide whether the graph has a path of length at least diam(G)+k(diam(G)denotes the length of a longest shortest path in a graph G). We obtain dichotomy results about Longest Path above Diameter on undirected and directed graphs. For (un)directed graphs, Longest Path above Diameter is NP-complete even for k=1. However, if the input undirected graph is 2-connected, then the problem is FPT. On the other hand, for 2-connected directed graphs, we show that Longest Path above Diameter is solvable in polynomial time for each k ∈ {1,..., 4} and is NP-complete for every k ≥ 5. The parameterized complexity of Longest Detour on general directed graphs remains an interesting open problem.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • longest path
  • longest detour
  • diameter
  • directed graphs
  • parameterized complexity

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