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

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

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • digraphs
  • parameterized complexity
  • disjoint paths
  • butterfly minors
  • immersions
  • ear anonymity

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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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.IPEC.2025.2

Author Details

Ken-ichi Kawarabayashi
  • National Institute of Informatics, Tokyo, Japan
Nicola Lorenz
  • Universität Hamburg, Germany
Marcelo Garlet Milani
  • National Institute of Informatics, Tokyo, Japan
Jacob Stegemann
  • Universität Hamburg, Germany

Funding

Research of Ken-ichi Kawarabayashi and Marcelo Garlet Milani supported by JSPS KAKENHI JP20A402 and 22H05001, and by JST ASPIRE JPMJAP2302.

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