  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.
