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. In this paper, we study the approximability of the problem on a large variety of graph classes. Our first result is a linear-time $(1+\varepsilon)$-approximation 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 directly yields linear-size $(1+\varepsilon)$-approximation lossy kernels for the problems on any bounded-expansion graph classes. Our second result is a linear-time approximation scheme for (Induced) Subgraph Hitting on any graph class $\mathcal{G}$ of polynomial expansion, based on the local-search framework of Har-Peled and Quanrud [SICOMP 2017]. This approximation scheme can be applied to a more general family of problems that aim to hit all subgraphs satisfying a certain property $\pi$ that is efficiently testable and has bounded diameter. Both of our results have applications to Subgraph Hitting (not induced) on wide classes of geometric intersection graphs, resulting in linear-size lossy kernels and (near-)linear time approximation schemes for the problem.

翻译：我们研究一个基础的顶点删除问题——（诱导）子图击中问题：给定图$G$和禁止图集$\mathcal{F}$，目标是找到$G$的最小顶点子集$S$，使得$G-S$不包含$\mathcal{F}$中任何图作为（诱导）子图。这是一个通用问题，涵盖了许多被广泛独立研究的著名问题，特别是（但不限于）从近似和参数化两个角度。本文研究了该问题在多种图类上的可近似性。我们的第一个结果是：对于任意有界扩张图类$\mathcal{G}$，可以将（诱导）子图击中问题以线性时间$(1+\varepsilon)$-近似归约到$\mathcal{G}$内有界度图上的同一问题。这直接为任意有界扩张图类上的问题提供了线性规模的$(1+\varepsilon)$-近似有损核。第二个结果是：基于Har-Peled和Quanrud [SICOMP 2017]的局部搜索框架，为任意多项式扩张图类$\mathcal{G}$上的（诱导）子图击中问题设计了线性时间近似方案。该近似方案可适用于更广泛的问题族，这些问题的目标是对所有满足特定性质$\pi$的子图进行删除，其中$\pi$需具有高效可测试性和有界直径。我们的两个结果在广泛的几何交图类上对（非诱导）子图击中问题均有应用，为该问题实现了线性规模的有损核与（近）线性时间近似方案。