Let $S_1,S_2,\ldots,S_n$ be sets that may have elements in common. I'm looking for a smallest set $X$ such that $\forall i,\,X\cap S_i \ne \emptyset$.
Does this problem have a name? Or does it reduce to some known problem?
In my context $S_1,\ldots,S_n$ describe the elementary cycles of a strongly connected component, and I'm looking for a smallest set of vertices $X$ that intersects all cycles.