Hardness of the Generalized Coloring Numbers

The generalized coloring numbers of Kierstead and Yang [7] offer an algorithmically useful characterization of graph classes with bounded expansion. In this work, we consider the hardness and approximability of these parameters. First, we show that it is NP-hard to compute the weak 2-coloring number (answering an open question of Grohe et al. [5]). We then complete the picture by proving that the $r$-coloring number is also NP-hard to compute for all $r \geq 2$. Finally, we give an approximation algorithm for the $r$-coloring number which improves both the runtime and approximation factor of the existing approach of Dvo\v{r}\'ak [3]. Our algorithm greedily orders vertices with small enough $i$-reach for every $i \leq r$ and achieves an $O(C_{r-1} k^{r-1})$-approximation, where $C_j$ is the $j$th Catalan number.