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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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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).

References

  1. Saeed A. Amiri, Stephan Kreutzer, Dániel Marx, and Roman Rabinovich. Routing with congestion in acyclic digraphs. Information Processing Letters, 151:105836, 2019. URL: https://doi.org/10.1016/j.ipl.2019.105836.
  2. Jørgen Bang-Jensen and Caresten Thomassen. Personal communication. Personal Communication, 2024. Google Scholar
  3. Jørgen Bang-Jensen and Carsten Thomassen. A polynomial algorithm for the 2-path problem for semicomplete digraphs. SIAM Journal on Discrete Mathematics, 5(3):366-376, 1992. URL: https://doi.org/10.1137/0405027.
  4. Jørgen Bang-Jensen and Gregory Gutin. Classes of Directed Graphs. Springer Monographs in Mathematics, 2018. URL: https://doi.org/10.1007/978-3-319-71840-8.
  5. Florian Barbero, Christophe Paul, and Michal Pilipczuk. Strong immersion is a well-quasi-ordering for semicomplete digraphs. Journal of Graph Theory, 90(4):484-496, 2019. URL: https://doi.org/10.1002/JGT.22408.
  6. Adrian Bondy and U.S.R. Murty. Graph Theory. Springer-Verlag London, 2008. Google Scholar
  7. Ulrik Brandes, Wolfram Schlickenrieder, Gabriele Neyer, Dorothea Wagner, and Karsten Weihe. A software package of algorithms and heuristics for disjoint paths in planar networks. Discrete Applied Mathematics, 92(2-3):91-110, 1999. URL: https://doi.org/10.1016/S0166-218X(99)00048-7.
  8. Rowland Leonard Brooks. On colouring the nodes of a network. Mathematical Proceedings of the Cambridge Philosophical Society, 37(2):194-197, 1941. URL: https://doi.org/10.1017/S030500410002168X.
  9. Victor Campos, Raul Lopes, Ana Karolinna Maia, and Ignasi Sau. Adapting the Directed Grid Theorem into an FPT Algorithm. SIAM Journal on Discrete Mathematics, 36(3):1887-1917, 2022. URL: https://doi.org/10.1137/21M1452664.
  10. Victor A. Campos, Jonas Costa, Raul Lopes, and Ignasi Sau. New Menger-Like Dualities in Digraphs and Applications to Half-Integral Linkages. In Proc. of the 31st Annual European Symposium on Algorithms (ESA), volume 274 of LIPIcs, pages 30:1-30:18, 2023. URL: https://doi.org/10.4230/LIPICS.ESA.2023.30.
  11. Dario Giuliano Cavallaro, Ken-ichi Kawarabayashi, and Stephan Kreutzer. Edge-Disjoint Paths in Eulerian Digraphs. In Proc. of the 56th Annual ACM Symposium on Theory of Computing (STOC), pages 704-715, 2024. URL: https://doi.org/10.1145/3618260.3649758.
  12. Chandra Chekuri, Sanjeev Khanna, and F. Bruce Shepherd. An o(√n) approximation and integrality gap for disjoint paths and unsplittable flow. Theory of Computing, 2(7):137-146, 2006. URL: https://doi.org/10.4086/TOC.2006.V002A007.
  13. Maria Chudnovsky, Alexandra O. Fradkin, and Paul D. Seymour. Tournament immersion and cutwidth. Journal of Combinatorial Theory, Series B, 102(1):93-101, 2012. URL: https://doi.org/10.1016/J.JCTB.2011.05.001.
  14. Maria Chudnovsky, Alex Scott, and Paul Seymour. Disjoint paths in unions of tournaments. Journal of Combinatorial Theory, Series B, 135:238-255, 2019. URL: https://doi.org/10.1016/j.jctb.2018.08.007.
  15. Maria Chudnovsky and Paul D. Seymour. A well-quasi-order for tournaments. Journal of Combinatorial Theory, Series B, 101(1):47-53, 2011. URL: https://doi.org/10.1016/J.JCTB.2010.10.003.
  16. Maria Chudnovsky, Paul D. Seymour, and Alex Scott. Disjoint paths in tournaments. Advances in Mathematics, 270:582-597, 2015. URL: https://doi.org/10.1016/j.aim.2014.11.011.
  17. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  18. Andreas Darmann and Janosch Döcker. On simplified np-complete variants of monotone 3 -sat. Discrete Applied Mathematics, 292:45-58, 2021. URL: https://doi.org/10.1016/j.dam.2020.12.010.
  19. Rod G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. Google Scholar
  20. Katherine Edwards, Irene Muzi, and Paul Wollan. Half-Integral Linkages in Highly Connected Directed Graphs. In Proc. of the 25th Annual European Symposium on Algorithms (ESA), volume 87 of LIPIcs, pages 36:1-36:12, 2017. Full version available in https://arxiv.org/abs/1611.01004. URL: https://doi.org/10.4230/LIPIcs.ESA.2017.36.
  21. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Springer-Verlag, 2006. URL: https://doi.org/10.1007/3-540-29953-X.
  22. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, 2019. URL: https://doi.org/10.1017/9781107415157.
  23. Fedor V. Fomin and Michal Pilipczuk. On width measures and topological problems on semi-complete digraphs. Journal of Combinatorial Theory, Series B, 138:78-165, 2019. URL: https://doi.org/10.1016/J.JCTB.2019.01.006.
  24. Fedor V. Fomin and Michał Pilipczuk. On width measures and topological problems on semi-complete digraphs. Journal of Combinatorial Theory, Series B, 138:78-165, 2019. URL: https://doi.org/10.1016/j.jctb.2019.01.006.
  25. Steven Fortune, John E. Hopcroft, and James Wyllie. The directed subgraph homeomorphism problem. Theoretical Computer Science, 10:111-121, 1980. URL: https://doi.org/10.1016/0304-3975(80)90009-2.
  26. Alexandra O. Fradkin and Paul D. Seymour. Tournament pathwidth and topological containment. Journal of Combinatorial Theory, Series B, 103(3):374-384, 2013. URL: https://doi.org/10.1016/J.JCTB.2013.03.001.
  27. Alexandra O. Fradkin and Paul D. Seymour. Edge-disjoint paths in digraphs with bounded independence number. Journal of Combinatorial Theory, Series B, 110:19-46, 2015. URL: https://doi.org/10.1016/J.JCTB.2014.07.002.
  28. Robert Ganian, Petr Hliněný, Joachim Kneis, Alexander Langer, Jan Obdržálek, and Peter Rossmanith. Digraph width measures in parameterized algorithmics. Discrete Applied Mathematics, 168:88-107, 2014. URL: https://doi.org/10.1016/j.dam.2013.10.038.
  29. Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. URL: https://dl.acm.org/doi/10.5555/574848.
  30. Archontia C. Giannopoulou, Ken-ichi Kawarabayashi, Stephan Kreutzer, and O-joung Kwon. The canonical directed tree decomposition and its applications to the directed disjoint paths problem, 2020. URL: https://arxiv.org/abs/2009.13184.
  31. Guilherme C. M. Gomes, Raul Lopes, and Ignasi Sau. Revisiting directed disjoint paths on tournaments (and relatives), 2025. URL: https://arxiv.org/abs/2504.19957.
  32. Meike Hatzel, Stephan Kreutzer, Marcelo Garlet Milani, and Irene Muzi. Cycles of Well-Linked Sets and an Elementary Bound for the Directed Grid Theorem. In Proc. of the 65th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 1-20, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00011.
  33. Ken-ichi Kawarabayashi, Yusuke Kobayashi, and Stephan Kreutzer. An excluded half-integral grid theorem for digraphs and the directed disjoint paths problem. In Proc. of the 46th Annual ACM on Symposium on Theory of Computing (STOC), pages 70-78, 2014. URL: https://doi.org/10.1145/2591796.2591876.
  34. Ken-ichi Kawarabayashi and Stephan Kreutzer. The Directed Grid Theorem. In Proc. of the 47th Annual ACM on Symposium on Theory of Computing (STOC), pages 655-664. ACM, 2015. URL: https://doi.org/10.1145/2746539.2746586.
  35. Ilhee Kim and Paul D. Seymour. Tournament minors. Journal of Combinatorial Theory, Series B, 112:138-153, 2015. URL: https://doi.org/10.1016/J.JCTB.2014.12.005.
  36. Kenta Kitsunai, Yasuaki Kobayashi, and Hisao Tamaki. On the pathwidth of almost semicomplete digraphs. In Nikhil Bansal and Irene Finocchi, editors, Proc. of the 23rd Annual European Symposium on Algorithms (ESA), volume 9294 of LNCS, pages 816-827, 2015. URL: https://doi.org/10.1007/978-3-662-48350-3_68.
  37. Raul Lopes and Ignasi Sau. A relaxation of the Directed Disjoint Paths problem: A global congestion metric helps. Theoretical Computer Science, 898:75-91, 2022. URL: https://doi.org/10.1016/j.tcs.2021.10.023.
  38. Lúcia Martins, Teresa Gomes, and David Tipper. Efficient heuristics for determining node-disjoint path pairs visiting specified nodes. Networks, 70(4):292-307, 2017. URL: https://doi.org/10.1002/NET.21778.
  39. Rolf Niedermeier. Invitation to Fixed-Parameter Algorithms. Oxford University Press, 2006. URL: https://doi.org/10.1093/acprof:oso/9780198566076.001.0001.
  40. Neil Robertson and Paul D. Seymour. Graph Minors. XIII. The Disjoint Paths Problem. Journal of Combinatorial Theory, Series B, 63(1):65-110, 1995. URL: https://doi.org/10.1006/JCTB.1995.1006.
  41. Aleksandrs Slivkins. Parameterized tractability of edge-disjoint paths on directed acyclic graphs. SIAM Journal on Discrete Mathematics, 24(1):146-157, 2010. URL: https://doi.org/10.1137/070697781.
  42. Carsten Thomassen. Highly connected non-2-linked digraphs. Combinatorica, 11(4):393-395, 1991. URL: https://doi.org/10.1007/BF01275674.
  43. Michal Wlodarczyk. Constant Approximating Disjoint Paths on Acyclic Digraphs Is W[1]-Hard. In Proc. of the 35th International Symposium on Algorithms and Computation (ISAAC), volume 322 of LIPIcs, pages 57:1-57:16, 2024. URL: https://doi.org/10.4230/LIPICS.ISAAC.2024.57.
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