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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.09868 |
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| _version_ | 1866918091023187968 |
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| author | Kawarabayashi, Ken-ichi Lorenz, Nicola Milani, Marcelo Garlet Stegemann, Jacob |
| author_facet | Kawarabayashi, Ken-ichi Lorenz, Nicola Milani, Marcelo Garlet Stegemann, Jacob |
| contents | 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 $c \geq 1$.
We strengthen the existing hardness result by showing that the problem remains W[1]-hard for every congestion $c \geq 1$ even if:
- the input digraph $D$ is acyclic,
- $D$ does not contain an acyclic $(5, 5)$-grid as a butterfly minor,
- $D$ does not contain an acyclic tournament on 9 vertices as a butterfly minor, and
- $D$ 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 $c \geq 1$ even if:
- the input digraph $D$ is acyclic,
- $D$ has maximum undirected degree 3,
- $D$ does not contain an acyclic $(7, 7)$-wall as a weak immersion and
- $D$ has ear-anonymity at most 5. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09868 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Directed disjoint paths remains W[1]-hard on acyclic digraphs without large grid minors Kawarabayashi, Ken-ichi Lorenz, Nicola Milani, Marcelo Garlet Stegemann, Jacob Computational Complexity 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 $c \geq 1$. We strengthen the existing hardness result by showing that the problem remains W[1]-hard for every congestion $c \geq 1$ even if: - the input digraph $D$ is acyclic, - $D$ does not contain an acyclic $(5, 5)$-grid as a butterfly minor, - $D$ does not contain an acyclic tournament on 9 vertices as a butterfly minor, and - $D$ 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 $c \geq 1$ even if: - the input digraph $D$ is acyclic, - $D$ has maximum undirected degree 3, - $D$ does not contain an acyclic $(7, 7)$-wall as a weak immersion and - $D$ has ear-anonymity at most 5. |
| title | Directed disjoint paths remains W[1]-hard on acyclic digraphs without large grid minors |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2507.09868 |