The notion of $\mathcal{H}$-treewidth, where $\mathcal{H}$ is a hereditary graph class, was recently introduced as a generalization of the treewidth of an undirected graph. Roughly speaking, a graph of $\mathcal{H}$-treewidth at most $k$ can be decomposed into (arbitrarily large) $\mathcal{H}$-subgraphs which interact only through vertex sets of size $O(k)$ which can be organized in a tree-like fashion. $\mathcal{H}$-treewidth can be used as a hybrid parameterization to develop fixed-parameter tractable algorithms for $\mathcal{H}$-deletion problems, which ask to find a minimum vertex set whose removal from a given graph $G$ turns it into a member of $\mathcal{H}$. The bottleneck in the current parameterized algorithms lies in the computation of suitable tree $\mathcal{H}$-decompositions. We present FPT approximation algorithms to compute tree $\mathcal{H}$-decompositions for hereditary and union-closed graph classes $\mathcal{H}$. Given a graph of $\mathcal{H}$-treewidth $k$, we can compute a 5-approximate tree $\mathcal{H}$-decomposition in time $f(O(k)) \cdot n^{O(1)}$ whenever $\mathcal{H}$-deletion parameterized by solution size can be solved in time $f(k) \cdot n^{O(1)}$ for some function $f(k) \geq 2^k$. The current-best algorithms either achieve an approximation factor of $k^{O(1)}$ or construct optimal decompositions while suffering from non-uniformity with unknown parameter dependence. Using these decompositions, we obtain algorithms solving Odd Cycle Transversal in time $2^{O(k)} \cdot n^{O(1)}$ parameterized by $\mathsf{bipartite}$-treewidth and Vertex Planarization in time $2^{O(k \log k)} \cdot n^{O(1)}$ parameterized by $\mathsf{planar}$-treewidth, showing that these can be as fast as the solution-size parameterizations and giving the first ETH-tight algorithms for parameterizations by hybrid width measures.

翻译：$\mathcal{H}$-树宽的概念（其中$\mathcal{H}$为遗传图类）近期被提出，作为无向图树宽的一种推广。粗略而言，$\mathcal{H}$-树宽至多为$k$的图可分解为（任意大的）$\mathcal{H}$-子图，这些子图仅通过规模为$O(k)$的顶点集相互作用，且该顶点集可组织成树状结构。$\mathcal{H}$-树宽可用作混合参数化方法，为$\mathcal{H}$-删除问题开发固定参数可解算法——这类问题要求寻找最小顶点集，使得从给定图$G$中移除该集合后所得图属于$\mathcal{H}$。当前参数化算法的瓶颈在于计算合适的树$\mathcal{H}$-分解。本文针对遗传且并封闭图类$\mathcal{H}$，提出FPT近似算法以计算树$\mathcal{H}$-分解。对于$\mathcal{H}$-树宽为$k$的图，若以解规模为参数的$\mathcal{H}$-删除问题可在$f(k) \cdot n^{O(1)}$时间内求解（其中$f(k) \geq 2^k$为某函数），则可在$f(O(k)) \cdot n^{O(1)}$时间内计算出5-近似的树$\mathcal{H}$-分解。当前最优算法或仅能达到$k^{O(1)}$的近似比，或需构建最优分解但存在非均匀性且参数依赖关系未知。利用这些分解，我们分别获得了以$\mathsf{二部}$-树宽为参数的奇圈横贯问题的$2^{O(k)} \cdot n^{O(1)}$时间算法，以及以$\mathsf{平面}$-树宽为参数的顶点平面化问题的$2^{O(k \log k)} \cdot n^{O(1)}$时间算法，表明其速度可与解规模参数化算法媲美，并首次给出针对混合宽度度量参数化的ETH紧致算法。