We revisit two well-studied problems, Bounded Degree Vertex Deletion and Defective Coloring, where the input is a graph $G$ and a target degree $\Delta$ and we are asked either to edit or partition the graph so that the maximum degree becomes bounded by $\Delta$. Both are known to be parameterized intractable for treewidth. We revisit the parameterization by treewidth, as well as several related parameters and present a more fine-grained picture of the complexity of both problems. Both admit straightforward DP algorithms with table sizes $(\Delta+2)^\mathrm{tw}$ and $(\chi_\mathrm{d}(\Delta+1))^{\mathrm{tw}}$ respectively, where tw is the input graph's treewidth and $\chi_\mathrm{d}$ the number of available colors. We show that both algorithms are optimal under SETH, even if we replace treewidth by pathwidth. Along the way, we also obtain an algorithm for Defective Coloring with complexity quasi-linear in the table size, thus settling the complexity of both problems for these parameters. We then consider the more restricted parameter tree-depth, and bridge the gap left by known lower bounds, by showing that neither problem can be solved in time $n^{o(\mathrm{td})}$ under ETH. In order to do so, we employ a recursive low tree-depth construction that may be of independent interest. Finally, we show that for both problems, an $\mathrm{vc}^{o(\mathrm{vc})}$ algorithm would violate ETH, thus already known algorithms are optimal. Our proof relies on a new application of the technique of $d$-detecting families introduced by Bonamy et al. Our results, although mostly negative in nature, paint a clear picture regarding the complexity of both problems in the landscape of parameterized complexity, since in all cases we provide essentially matching upper and lower bounds.

翻译：我们重新审视两个经典问题——有界度顶点删除与缺陷着色，其输入为图$G$和目标度$\Delta$，目标是通过编辑或分割图使得最大度不超过$\Delta$。已知这两个问题在树宽参数下是参数化难解的。我们重新考察树宽参数化及其相关参数，呈现两个问题复杂度的更精细图像。两者均存在直接动态规划算法，其表规模分别为$(\Delta+2)^\mathrm{tw}$和$(\chi_\mathrm{d}(\Delta+1))^{\mathrm{tw}}$，其中tw为输入图的树宽，$\chi_\mathrm{d}$为可用颜色数。我们证明在SETH假设下即使将树宽替换为路径宽，这两种算法均为最优。在此过程中，我们还获得一个缺陷着色算法，其复杂度在表规模上呈拟线性，从而确定了这两个参数下问题的复杂度。随后考虑更受限的树深参数，通过证明在ETH假设下两个问题均无法在$n^{o(\mathrm{td})}$时间内求解，填补了已知下界间的空白。为此，我们采用了递归的低树深构造方法，该方法可能具有独立研究价值。最终证明对于两个问题，$\mathrm{vc}^{o(\mathrm{vc})}$算法将违反ETH假设，因而已有算法已为最优。我们的证明依赖于Bonamy等人引入的$d$-检测族技术的新应用。尽管多数结论为否定性质，但我们的结果清晰描绘了参数化复杂度图景中这两个问题的复杂性——在所有情形下均提供了本质上匹配的上界与下界。