Hardness and Approximation for Coloring Digraphs

2026-05-19

AI Summary

This work investigates the approximability of the chromatic number and the size of the maximum acyclic subgraph in directed graphs, with a focus on special structures such as tournaments. Through complexity reductions and combinatorial graph-theoretic analysis, it establishes that both problems admit no polynomial-time approximation within a factor of $n^{1-\varepsilon}$ for any $\varepsilon > 0$, even when restricted to tournaments—matching the hardness known for undirected graphs. For $\ell$-colorable digraphs, the paper presents the first approximation algorithm that uses at most $\ell \cdot n^{1-1/\ell}$ colors and runs in $O(n^{2\ell})$ time. Notably, for 2-colorable digraphs, this yields an efficient polynomial-time algorithm employing only $2\sqrt{n}$ colors, substantially improving upon prior results.

Abstract

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.

Problem

• dichromatic number
• acyclic number
• digraph coloring
• approximation hardness
• tournaments

Innovation

• dichromatic number
• acyclic subdigraph
• approximation hardness
• directed graph coloring
• tournament