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：本文研究一类基本的顶点删除问题——（诱导）子图击中：给定图$G$和禁制图集$\mathcal{F}$，目标是计算$G$中顶点最小集$S$，使得$G-S$不包含$\mathcal{F}$中任一图作为（诱导）子图。这是一个通用问题，涵盖了许多独立研究已久的经典问题，尤其是在近似与参数化视角下。本文研究该问题在多种图类上的可近似性。第一项成果是在任意有界扩张图类$\mathcal{G}$上，将（诱导）子图击中问题以线性时间$(1+\varepsilon)$-近似归约到同一图类内度有界图上的同类问题。这直接为任意有界扩张图类上的该问题提供了线性规模的$(1+\varepsilon)$-近似亏损核。第二项成果是基于Har-Peled与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）子图击中问题设计线性时间近似方案。该近似方案可适用于更广泛的问题族，其目标为击中所满足特定性质$\pi$（该性质可高效检验且具有有界直径）的所有子图。两项成果均可应用于广泛几何交图类上的子图击中（非诱导）问题，从而为该问题提供线性规模亏损核与（近）线性时间近似方案。