Call a family of sets $\mathcal{F} = \{S_1, \dotsc, S_k\}$ diverse if each set $S_i \in \mathcal{F}$ has at least one unique element.  What are possible approaches for finding the largest diverse set $S$ in a family of sets $\mathcal{F}$?One approach is to solve a modified set packing problem.  Suppose $\mathcal{F}=\{S_1,\dotsc,S_k\}$. Let $K$ be a subset of elements, $K \subset \bigcup S_i$, and let $\mathcal{F}_{-K}=\{S_1 \setminus K,\dotsc, S_k \setminus K\}$.  Then the maximal diverse set $S$ corresponds to the largest maximal set packing obtained from $\mathcal{F}_{-L}$ where $L$ is the set of all non-unique elements in $\mathcal{F}$.But, what's a good heuristic for choosing $K$?  Or are there better approaches altogether?
Problem similar to set packing
algorithms;combinatorics;heuristics