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DOI: 10.4230/LIPIcs.IPEC.2025.2

Directed Disjoint Paths Remains W[1]-Hard on Acyclic Digraphs Without Large Grid Minors

Authors: Ken-ichi Kawarabayashi, Nicola Lorenz, Marcelo Garlet Milani, and Jacob Stegemann

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract

In the Vertex-Disjoint-Paths-With-Congestion problem, the input consists of a digraph D, an integer c and k pairs of vertices (s_i, t_i), and the task is to find a set of paths connecting each s_i to its corresponding t_i, whereas each vertex of D appears in at most c many paths. The case where c = 1 is known to be NP-complete even if k = 2 [Fortune, Hopcroft and Wyllie, 1980] on general digraphs and is W[1]-hard with respect to k (excluding the possibility of an f(k)n^O(1)-time algorithm under standard assumptions) on acyclic digraphs [Slivkins, 2010]. The proof of [Slivkins, 2010] can also be adapted to show W[1]-hardness with respect to k for every congestion c ≥ 1. We strengthen the existing hardness result by showing that the problem remains W[1]-hard for every congestion c ≥ 1 even if: (1) the input digraph D is acyclic, (2) D does not contain an acyclic (5, 5)-grid as a butterfly minor, (3) D does not contain an acyclic tournament on 9 vertices as a butterfly minor, and (4) D has ear-anonymity at most 5. Further, we also show that the edge-congestion variant of the problem remains W[1]-hard for every congestion c ≥ 1 even if: (1) the input digraph D is acyclic, (2) D has maximum undirected degree 3, (3) D does not contain an acyclic (7, 7)-wall as a weak immersion and (4) D has ear-anonymity at most 5.

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Ken-ichi Kawarabayashi, Nicola Lorenz, Marcelo Garlet Milani, and Jacob Stegemann. Directed Disjoint Paths Remains W[1]-Hard on Acyclic Digraphs Without Large Grid Minors. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 2:1-2:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kawarabayashi_et_al:LIPIcs.IPEC.2025.2,
  author =	{Kawarabayashi, Ken-ichi and Lorenz, Nicola and Garlet Milani, Marcelo and Stegemann, Jacob},
  title =	{{Directed Disjoint Paths Remains W\lbrack1\rbrack-Hard on Acyclic Digraphs Without Large Grid Minors}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{2:1--2:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.2},
  URN =		{urn:nbn:de:0030-drops-251347},
  doi =		{10.4230/LIPIcs.IPEC.2025.2},
  annote =	{Keywords: digraphs, parameterized complexity, disjoint paths, butterfly minors, immersions, ear anonymity}
}

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Directed Disjoint Paths Remains W[1]-Hard on Acyclic Digraphs Without Large Grid Minors

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