Maximize number of covered sets by choosing given number of elements I am not sure if it's a known probem:
There is a set of some elements. For the purpose of this explanation that can be a subset of natural numbers, let's say $\{1, 2, ..., 20\}$. Let's call it $SET$.
There are also given subsets of the $SET$. That subsets don't have to be disjoint and sum of them doesn't have to cover the $SET$ (i.e. $\{1, 3, 5\}$, $\{8, 9, 14\}$, $\{1, 10, 15, 18, 20\}$, $\{5, 6, 8, 9\}$, and so).
Now, with given number $k$ ($0<k<|SET|$), how to choose $k$ elements of the $SET$ to maximize number of covered (by choosen elements) subsets.
Thank you for help,
Przemek.

Could someone explain the way that Mees de Vries answer meets my criteria. I think it is about searching smallest number of subsets covering SET, not about finding vertices that satisfy the maximum number of subsets. Correct me if I am wrong.
