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Revisiting Directed Disjoint Paths on Tournaments (And Relatives)

Authors Guilherme de C. M. Gomes , Raul Lopes , Ignasi Sau

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LIPIcs.ICALP.2025.90.pdf
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ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Combinatorics
Keywords
  • directed graphs
  • tournaments
  • semicomplete digraphs
  • directed disjoint paths
  • congestion
  • parameterized complexity
  • directed pathwidth

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Abstract

In the Directed Disjoint Paths problem (k-DDP), we are given a digraph and k pairs of terminals, and the goal is to find k pairwise vertex-disjoint paths connecting each pair of terminals. Bang-Jensen and Thomassen [SIAM J. Discrete Math. 1992] claimed that k-DDP is NP-complete on tournaments, and this result triggered a very active line of research about the complexity of the problem on tournaments and natural superclasses. We identify a flaw in their proof, which has been acknowledged by the authors, and provide a new NP-completeness proof. From an algorithmic point of view, Fomin and Pilipczuk [J. Comb. Theory B 2019] provided an FPT algorithm for the edge-disjoint version of the problem on semicomplete digraphs, and showed that their technique cannot work for the vertex-disjoint version. We overcome this obstacle by showing that the version of k-DDP where we allow congestion c on the vertices is FPT on semicomplete digraphs provided that c is greater than k/2. This is based on a quite elaborate irrelevant vertex argument inspired by the edge-disjoint version, and we show that our choice of c is best possible for this technique, with a counterexample with no irrelevant vertices when c ≤ k/2. We also prove that k-DDP on digraphs that can be partitioned into h semicomplete digraphs is W[1]-hard parameterized by k+h, which shows that the XP algorithm presented by Chudnovsky, Scott, and Seymour [J. Comb. Theory B 2019] is essentially optimal.

Cite As Get BibTex

Guilherme de C. M. Gomes, Raul Lopes, and Ignasi Sau. Revisiting Directed Disjoint Paths on Tournaments (And Relatives). In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 90:1-90:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.90

Author Details

Guilherme de C. M. Gomes
  • LIRMM, Université de Montpellier, CNRS, Montpellier, France
  • Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
Raul Lopes
  • LIRMM, Université de Montpellier, CNRS, Montpellier, France
  • Hamburg University of Technology, Institute for Algorithms and Complexity, Hamburg, Germany
Ignasi Sau
  • LIRMM, Université de Montpellier, CNRS, Montpellier, France

Funding

  • de C. M. Gomes, Guilherme: Funded by the European Union, project PACKENUM, grant number 101109317. Views and opinions expressed are however those of the author only and do not necessarily reflect those of the European Union. Neither the European Union nor the granting authority can be held responsible for them.
  • Lopes, Raul: French project ELIT (ANR-20-CE48-0008-01) and HIDSS-0002 DASHH (Data Science in Hamburg - Helmholtz Graduate School for the Structure of Matter).
  • Sau, Ignasi: French project ELIT (ANR-20-CE48-0008-01).

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