Consider a set of distinct elements $\{a_1, a_2, a_3,\dots, a_n\}$ (the universe) and a collection of sets $S= \{ S_1, S_2, S_3,\dots, S_m \} $ with each set formed out of any strictly positive number of elements.
For a given number $k$ with $0<k < n$, find the $k$ elements that maximize the number of covered sets.
This problem is similar but not identical to the set cover problem and maximum coverage problem.
How is this problem called and which algorithm should be used to solve it?
For $1 \leq i \leq n$, let $B_i = \{S_j: S_j \in S, x_i \in S_j\}$. Then apply the maximum coverage problem to the sets $B_i$.