We investigate a fundamental vertex-deletion problem called (Induced) Subgraph Hitting: given a graph $G$ and a set $\mathcal{F}$ of forbidden graphs, the goal is to compute a minimum-sized set $S$ of vertices of $G$ such that $G-S$ does not contain any graph in $\mathcal{F}$ as an (induced) subgraph. This is a generic problem that encompasses many well-known problems that were extensively studied on their own, particularly (but not only) from the perspectives of both approximation and parameterization. We focus on the design of efficient approximation schemes, i.e., with running time $f(\varepsilon,\mathcal{F}) \cdot n^{O(1)}$, which are also of significant interest to both communities. Technically, our main contribution is a linear-time approximation-preserving reduction from (Induced) Subgraph Hitting on any graph class $\mathcal{G}$ of bounded expansion to the same problem on bounded degree graphs within $\mathcal{G}$. This yields a novel algorithmic technique to design (efficient) approximation schemes for the problem on very broad graph classes, well beyond the state-of-the-art. Specifically, applying this reduction, we derive approximation schemes with (almost) linear running time for the problem on any graph classes that have strongly sublinear separators and many important classes of geometric intersection graphs (such as fat-object graphs, pseudo-disk graphs, etc.). Our proofs introduce novel concepts and combinatorial observations that may be of independent interest (and, which we believe, will find other uses) for studies of approximation algorithms, parameterized complexity, sparse graph classes, and geometric intersection graphs. As a byproduct, we also obtain the first robust algorithm for $k$-Subgraph Isomorphism on intersection graphs of fat objects and pseudo-disks, with running time $f(k) \cdot n \log n + O(m)$.

翻译：我们研究一个名为（导出）子图命中的基本顶点删除问题：给定图 $G$ 和一个禁止图集合 $\mathcal{F}$，目标是计算 $G$ 的最小顶点子集 $S$，使得 $G-S$ 不包含 $\mathcal{F}$ 中任何图作为（导出）子图。这是一个通用问题，涵盖了许多已广泛研究的具体问题（特别但不限于近似和参数化两个视角）。我们专注于设计高效的近似方案，即运行时间为 $f(\varepsilon,\mathcal{F}) \cdot n^{O(1)}$ 的方案，这类方案对两个领域的研究群体均具有重要价值。技术层面上，我们的主要贡献是一个线性时间的近似保持归约方法，可将任意有界扩展图类 $\mathcal{G}$ 上的（导出）子图命中问题归约为该图类中有界度图上的相同问题。这提供了一种新颖的算法技术，能够在远超现有技术水平的极其广泛的图类上设计该问题的（高效）近似方案。具体而言，应用此归约，我们在所有具有强亚线性分隔子的图类以及众多重要的几何交图类（如胖体图、伪圆盘图等）上，获得了（几乎）线性运行时间的近似方案。我们的证明引入了新颖的概念和组合观察，这些内容可能对近似算法、参数化复杂性、稀疏图类和几何交图的研究具有独立价值（并且我们相信将在其他领域找到应用）。作为副产品，我们还在胖体和伪圆盘的交图上首次获得了 $k$-子图同构问题的鲁棒算法，其运行时间为 $f(k) \cdot n \log n + O(m)$。