# Minimun k-union from a different angle

I'm looking for work done on solving some problem which is very similar to the minimum k-union.

The problem:

There's a set of elements $$E=\{e_1,e_2,...,e_k\}$$ of size $$k$$, and a family of sets $$S_1,S_2,...,S_t$$ where each set $$S_i$$ is contained in $$E$$: $$S_i \subseteq E$$.

The objective is to find for each $$1 \leq j\leq k$$ the "best" subset of $$E$$ of size $$j$$, where "best" means that it has the biggest number of sets $$S_i$$ which are contained in it. The problem, as well as it being NP-hard, is described here too.

I could use solutions to the minimum k-union to solve this, but that wouldn't be ideal regarding computation time, so hopefully this problem can be reduced to some problem that has approximation algorithms to it.