The generalized coloring numbers of Kierstead and Yang (Order 2003) 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. (WG 2015) by showing that computing the weak 2-coloring number is NP-hard. Our approach further establishes that determining if a graph has weak $r$-coloring number at most $k$ is para-NP-hard when parameterized by $k$ for all $r \geq 2$. We adapt this to determining if a graph has $r$-coloring number at most $k$ as well, proving para-NP-hardness for all $r \geq 2$. Para-NP-hardness implies that no XP algorithm (runtime $O(n^{f(k)})$) exists for testing if a generalized coloring number is at most $k$. Moreover, there exists a constant $c$ such that it is NP-hard to approximate the generalized coloring numbers within a factor of $c$. To complement these results, we give an approximation algorithm for the generalized coloring numbers, improving both the runtime and approximation factor of the existing approach of Dvo\v{r}\'{a}k (EuJC 2013). We prove that greedily ordering vertices with small estimated backconnectivity achieves a $(k-1)^{r-1}$-approximation for the $r$-coloring number and an $O(k^{r-1})$-approximation for the weak $r$-coloring number.

翻译：Kierstead和Yang（Order 2003）提出的广义着色数为具有有界扩张的图类提供了算法上有用的刻画。本文研究了这些参数的难解性与可近似性。首先，我们完善了Grohe等人（WG 2015）的工作，证明计算弱2-着色数是NP困难的。我们的方法进一步表明，对于所有$r \geq 2$，判定图是否具有至多$k$的弱$r$-着色数在以$k$为参数化时是para-NP困难的。我们将这一结果推广至判定图是否具有至多$k$的$r$-着色数，证明了对于所有$r \geq 2$的情形均为para-NP困难。para-NP困难性意味着不存在XP算法（运行时间$O(n^{f(k)})$）用于测试广义着色数是否至多为$k$。此外，存在常数$c$使得在$c$因子内近似广义着色数是NP困难的。为补充这些结论，我们给出了广义着色数的近似算法，改进了Dvořák（EuJC 2013）现有方法在运行时间和近似因子方面的性能。我们证明，通过贪心排序具有较小估计后向连通性的顶点，可实现$r$-着色数的$(k-1)^{r-1}$近似和弱$r$-着色数的$O(k^{r-1})$近似。