Given a set of sets, find the magnitude (number of elements) of the smallest set containing at least one element from each set

Question

I know that the hitting problem is NP hard, but is it possible to find the magnitude of the smallest set? Also, provide the runtime.

Answer 1

When we say that hitting set is NP-hard, we mean that the following decision problem is NP-hard: given a set system $S$ and an integer $k$, decide whether there is a hitting set for $S$ of size at most $k$.