Let $\pi$ be a property of pairs $(G,Z)$, where $G$ is a graph and $Z\subseteq V(G)$. In the \emph{minimum $\pi$-hitting set problem}, given an input graph $G$, we want to find a smallest set $X\subseteq V(G)$ such that $X$ intersects every set $Z\subseteq V(G)$ such that $(G,Z)$ has the property $\pi$. An important special case is that $\pi$ is satisfied by $(G,Z)$ exactly if $G[Z]$ is isomorphic to one of graphs in a finite set $\mathcal{F}$; in this \emph{minimum $\mathcal{F}$-hitting set} problem, $X$ needs to hit all appearances of the graphs from $\mathcal{F}$ as induced subgraphs of $G$. In this note, we show that the local search argument of Har-Peled and Quanrud gives a PTAS for the minimum $\mathcal{F}$-hitting set problem for graphs from any class with polynomial expansion. Moreover, we argue that the local search argument applies more generally to all properties $\pi$ such that one can test whether $X$ is a $\pi$-hitting set in polynomial time and $G[Z]$ has bounded diameter whenever $(G,Z)$ satisfies $\pi$; this is a common generalization of the minimum $\mathcal{F}$-hitting set problem and minimum $r$-dominating set problem. Finaly, we note that the analogous claim also holds for the dual problem of finding the maximum number of disjoint sets $Z$ such that $(G,Z)$ has the property $\pi$; this generalizes maximum $F$-matching, maximum induced $F$-matching, and maximum $r$-independent set problems.

翻译：设 $\pi$ 为二元组 $(G,Z)$ 的一个性质，其中 $G$ 是一个图，$Z\subseteq V(G)$。在 \emph{最小 $\pi$-击打集问题} 中，给定输入图 $G$，我们需找到最小子集 $X\subseteq V(G)$，使得 $X$ 与每个满足性质 $\pi$ 的 $(G,Z)$ 中的集合 $Z\subseteq V(G)$ 均有交集。一个重要特例是：当且仅当 $G[Z]$ 同构于有限集 $\mathcal{F}$ 中的某个图时，$(G,Z)$ 满足 $\pi$；在此 \emph{最小 $\mathcal{F}$-击打集问题} 中，$X$ 需击打 $\mathcal{F}$ 中所有作为 $G$ 的导出子图出现的图。本文指出，Har-Peled 和 Quanrud 的局部搜索论证可为任意多项式扩张图类上的最小 $\mathcal{F}$-击打集问题提供多项式时间近似方案（PTAS）。进一步，我们论证该局部搜索论证更一般地适用于所有满足以下条件的性质 $\pi$：可在多项式时间内检验 $X$ 是否为 $\pi$-击打集，且当 $(G,Z)$ 满足 $\pi$ 时 $G[Z]$ 直径有界；这是最小 $\mathcal{F}$-击打集问题与最小 $r$-支配集问题的共同推广。最后，我们指出类似结论也适用于对偶问题——寻找满足性质 $\pi$ 的 $(G,Z)$ 中互不相交集合 $Z$ 的最大数量；该问题推广了最大 $F$-匹配、最大导出 $F$-匹配及最大 $r$-独立集问题。