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.
翻译:在 $k 的颜色中, 我们被赋予了一种“ 最接近 ” 的参数, 并且被要求将它的脊椎分隔在 $ k 的 参数中, 所以每套设置会引出一个 DAG 。 这个众所周知的问题是 NP 硬的, 因为它一般化( 非方向 ) $k美元 的颜色, 但是如果输入分解是周期性的, 就会变得微不足道。 这提出了自然参数化的复杂问题。 在本文中, 我们使用测量输入距离到 美元周期的参数, 以直接或非方向的 美元计数来计算。 因此, 几乎每套设置一个 $ 2, $ k 的颜色颜色是 NP, 最多是 $ 美元 。 我们用 $k 的颜色表示这个比例是 4 美元 。