The dichromatic number $\vecχ(D)$ of a digraph is the minimum number $k$ such that $V(D)$ can be partitioned into $k$ subsets, each inducing an acyclic digraph. The acyclic number $\vecα(D)$ is the cardinality of a largest induced acyclic subdigraph of $D$. We study these problems from an approximation point of view. We begin with establishing that even when restricted to tournaments, approximating $\vecχ$ and $\vecα$ remain as challenging as their undirected counterparts on general graphs. Specifically, we establish that for every $ε>0$, it is hard to approximate both $\vecα$ and $\vecχ$ up to a factor of $n^{1-ε}$ even when restricted to tournaments. We next consider approximate coloring of digraphs in special cases. We begin with establishing that we can color $\ell$-dicolorable digraphs using at most $\ell \cdot n^{1-\frac{1}{\ell}}$ colors in time $O(n^{2\ell})$; in particular, we can color $2$-dicolorable digraphs with $2\sqrt{n}$ colors in polynomial time. We then focus on bounding the dichromatic number of dense digraphs as a function of the independence number $α$ of the underlying graph. We consider two special cases in this regard: digraphs with $\vecχ(D)\leq 2$ and digraphs that do not contain any directed triangle. For these cases, we present algorithms which generalize and improve existing tools and results.

翻译：有向图 $D$ 的二色数 $\vecχ(D)$ 是将顶点集 $V(D)$ 划分为 $k$ 个无环有向子集所需的最小整数 $k$，其中每个子集诱导的子图均为无环有向图。有向无环数 $\vecα(D)$ 是 $D$ 中最大诱导无环有向子图的基数。我们从近似算法的角度研究这些问题。首先证明，即使限制在竞赛图上，$\vecχ$ 与 $\vecα$ 的近似问题仍与一般无向图上的对应问题难度相当。具体而言，我们证明：对任意 $ε>0$，即使限制在竞赛图上，$\vecα$ 与 $\vecχ$ 均难以在 $n^{1-ε}$ 因子内近似。其次考虑特殊情形下的有向图近似染色问题。首先证明：可在 $O(n^{2\ell})$ 时间内用至多 $\ell \cdot n^{1-\frac{1}{\ell}}$ 种颜色对 $\ell$ 可二色有向图进行染色；特别地，可在多项式时间内用 $2\sqrt{n}$ 种颜色对 $2$ 可二色有向图染色。随后，我们聚焦于稠密有向图的二色数关于基础图独立数 $α$ 的界。在此框架下考虑两类特殊情形：满足 $\vecχ(D)\leq 2$ 的有向图以及不含任何有向三角形的有向图。针对这些情形，我们提出能够推广并改进现有工具与结果的算法。