Directed disjoint paths remains W[1]-hard on acyclic digraphs without large grid minors

TL;DR

This paper proves that the Directed Disjoint Paths problem remains W[1]-hard on acyclic digraphs even under various structural restrictions, highlighting the problem's computational difficulty despite simplified graph conditions.

Contribution

The authors strengthen existing hardness results by showing W[1]-hardness persists on acyclic digraphs with several structural constraints, including absence of large grid minors and bounded ear-anonymity.

Findings

W[1]-hardness holds for all congestion levels c ≥ 1 on acyclic digraphs.

Hardness persists even when the graph excludes large grid minors and tournaments.

Edge-congestion variant remains W[1]-hard under degree and minor exclusion constraints.

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 $c \geq 1$. 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.