Directed disjoint paths remains W1-hard on acyclic digraphs without large grid minors

In the Vertex Disjoint Paths with Congestion problem, the input consists of a digraph D, an integer c and k pairs of vertices (sᵢ, tᵢ), and the task is to find a set of paths connecting each sᵢ to its corresponding tᵢ, 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 W1-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 W1-hardness with respect to k for every congestion c 1. We strengthen the existing hardness result by showing that the problem remains W1-hard for every congestion c 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 W1-hard for every congestion c 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.