Hardness and Approximation for Coloring Digraphs

The dichromatic number $\vec\chi(D)$ 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 $\vec\alpha(D)$ 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 $\vec\chi$ and $\vec\alpha$ remain as challenging as their undirected counterparts on general graphs. Specifically, we establish that for every $\epsilon >0$, it is hard to approximate both $\vec\alpha$ and $\vec\chi$ up to a factor of $n^{1-\epsilon}$ 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 $\alpha$ of the underlying graph. We consider two special cases in this regard: digraphs with $\vec\chi(D)\leq 2$ and digraphs that do not contain any directed triangle. For these cases, we present algorithms which generalize and improve existing tools and results.