标题： 有向不相交路径问题在没有大网格子图的无环有向图上仍然是W[1]-难的 显示英文标题

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

摘要： 在顶点不相交路径与拥塞问题中，输入包括一个有向图$D$，一个整数$c$和$k$对顶点$(s_i, t_i)$，任务是找到一组路径，将每个$s_i$连接到其对应的$t_i$，而$D$中的每个顶点最多出现在$c$条路径中。 当$c = 1$已知在一般有向图上即使$k = 2$ [Fortune, Hopcroft 和 Wyllie, 1980] 也是 NP 完全的，并且在无环有向图上相对于$k$是 W[1]-难的（在标准假设下排除了存在$f(k)n^{O(1)}$-时间算法的可能性）[Slivkins, 2010]。[Slivkins, 2010] 的证明也可以进行调整，以显示对于每个拥塞$c \geq 1$，相对于$k$是 W[1]-难的。 我们通过证明即使满足以下条件，该问题对于每个拥塞$c \geq 1$仍然保持 W[1]-hard： - 输入有向图$D$是无环的， -$D$不包含无环的$(5, 5)$-grid 作为蝴蝶次要图， -$D$不包含9个顶点的无环竞赛图作为蝴蝶次要图，并且 -$D$的耳匿名性至多为5。 此外，我们还证明了该问题的边拥挤变体对于每个拥挤$c \geq 1$仍然是 W[1]-hard，即使： - 输入有向图$D$是无环的， -$D$的最大无向度为 3， -$D$不包含无环$(7, 7)$-墙作为弱浸入，并且 -$D$的耳匿名性至多为 5。 显示英文摘要

摘要： 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.