Smallest set that intersects some given sets 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.
Your first problem is the hypergraph transversal problem (aka the HITTING SET problem). The second problem is the FEEDBACK VERTEX&nbsp;SET problem. Both the problems are NP-complete.