Given a set of sets $S = \{S_1, S_2, \dots, S_n$}, let $S^{'} \subset S$ be the largest subset of S that obeys $\left| \bigcup_{S_i \in S^{'}}{S_i} \right| \leq k$. What is the complexity of finding $P = \bigcup_{S_i \in S^{'}}{S_i}$ ?
I have a feeling that this problem is NP-hard as it is very similar to the set cover problem, but I have been unable to prove it.