In $k$-Digraph Coloring we are given a digraph and are asked to partition its vertices into at most $k$ sets, so that each set induces a DAG. This well-known problem is NP-hard, as it generalizes (undirected) $k$-Coloring, but becomes trivial if the input digraph is acyclic. This poses the natural parameterized complexity question what happens when the input is "almost" acyclic. In this paper we study this question using parameters that measure the input's distance to acyclicity in either the directed or the undirected sense. It is already known that, for all $k\ge 2$, $k$-Digraph Coloring is NP-hard on digraphs of DFVS at most $k+4$. We strengthen this result to show that, for all $k\ge 2$, $k$-Digraph Coloring is NP-hard for DFVS $k$. Refining our reduction we obtain two further consequences: (i) for all $k\ge 2$, $k$-Digraph Coloring is NP-hard for graphs of feedback arc set (FAS) at most $k^2$; interestingly, this leads to a dichotomy, as we show that the problem is FPT by $k$ if FAS is at most $k^2-1$; (ii) $k$-Digraph Coloring is NP-hard for graphs of DFVS $k$, even if the maximum degree $\Delta$ is at most $4k-1$; we show that this is also almost tight, as the problem becomes FPT for DFVS $k$ and $\Delta\le 4k-3$. We then consider parameters that measure the distance from acyclicity of the underlying graph. We show that $k$-Digraph Coloring admits an FPT algorithm parameterized by treewidth, whose parameter dependence is $(tw!)k^{tw}$. Then, we pose the question of whether the $tw!$ factor can be eliminated. Our main contribution in this part is to settle this question in the negative and show that our algorithm is essentially optimal, even for the much more restricted parameter treedepth and for $k=2$. Specifically, we show that an FPT algorithm solving $2$-Digraph Coloring with dependence $td^{o(td)}$ would contradict the ETH.

中文翻译：

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有向图着色和非循环距离

在 $k$-Digraph Coloring 中，我们得到一个有向图，并被要求将它的顶点划分为最多 $k$ 个集合，这样每个集合都会产生一个 DAG。这个众所周知的问题是 NP-hard 问题，因为它概括了（无向）$k$-Coloring，但如果输入有向图是无环的，则变得微不足道。这提出了自然参数化复杂性问题，当输入“几乎”是非循环时会发生什么。在本文中，我们使用参数来研究这个问题，这些参数在有向或无向意义上测量输入到非循环的距离。众所周知，对于所有 $k\ge 2$，$k$-Digraph Coloring 在 DFVS 的有向图中最多为 $k+4$ 是 NP-hard 的。我们加强这个结果以表明，对于所有 $k\ge 2$，$k$-Digraph 着色对于 DFVS $k$ 是 NP-hard 的。改进我们的减少我们得到两个进一步的结果：（i）对于所有 $k\ge 2$，$k$-Digraph 着色对于最多 $k^2$ 的反馈弧集 (FAS) 图来说是 NP 难的；有趣的是，这导致了二分法，因为我们表明，如果 FAS 最多为 $k^2-1$，则问题是 $k$ 的 FPT；(ii) $k$-Digraph 着色对于 DFVS $k$ 的图是 NP-hard 的，即使最大度数 $\Delta$ 最多为 $4k-1$；我们表明这也几乎是紧的，因为问题变成了 DFVS $k$ 和 $\Delta\le 4k-3$ 的 FPT。然后我们考虑测量与底层图的非循环性的距离的参数。我们证明 $k$-Digraph Coloring 承认一个由 treewidth 参数化的 FPT 算法，其参数依赖是 $(tw!)k^{tw}$。然后，我们提出了是否可以消除 $tw!$ 因素的问题。我们在这部分的主要贡献是否定这个问题，并表明我们的算法本质上是最优的，即使对于更受限制的参数 treedepth 和 $k=2$。具体来说，我们表明，解决 $2$-Digraph 着色的 FPT 算法依赖于 $td^{o(td)}$ 将与 ETH 相矛盾。