Let $X$ be a ground set, and consider a collection $\mathscr{S}$ of subsets of $X$, $\mathscr{S} = \{S_1, \dots, S_n\}$.
We would like to find a collection $\mathscr{S}'$ with the property that for all $S \in \mathscr{S}$, there is some $S' \in \mathscr{S}'$ such that $S' \subseteq S$. We might also want $\mathscr{S}'$ to satisfy additional properties, such as that elements of $\mathscr{S}'$ have a certain minimum cardinality.
Is there any name for this type of design? It is similar to the set cover problem or hitting-set problems, but doesn't seem to fit exactly. Any references would be appreciated.