Hardness of the Generalized Coloring Numbers

The generalized coloring numbers of Kierstead and Yang 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 complete the work of Grohe et al. by showing that computing the weak 2-coloring number is NP-hard. Our approach further establishes that determining the weak $r$-coloring number is APX-hard for all $r \geq 2$. We adapt this to the $r$-coloring number as well, proving APX-hardness for all $r \geq 2$. Our reductions also imply that for every fixed $r \geq 2$, no XP algorithm (runtime $O(n^{f(k)})$) exists for testing if either generalized coloring number is at most $k$. 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}\'{a}k. Our algorithm greedily orders vertices with small enough $\ell$-reach for every $\ell \leq r$ and achieves an $O(C_{r-1} k^{r-1})$-approximation, where $C_i$ is the $i$th Catalan number.