A note on local search for hitting sets

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.