given a universe $\mathcal{U}$ of elements and a system $\mathcal{S}$ of weighted subsets of $\mathcal{U}$ whose union covers $\mathcal{U}$.


Assuming the existence of at least one subsytem $S\subseteq\mathcal{S}$ such that the disjoint union of its elements $\lbrace s\_1,\,\dots,\,s\_k\rbrace$ covers $\mathcal{U}$, what is the complexity of finding $S\_{opt}\subseteq\mathcal{S}$, that resembles the disjoint set-cover of minimal weightsum, i.e. does the restriction to covers with disjoint sets suffice to make the problem polynomial?



Complexity of set-partition problems
The [exact cover problem](https://en.wikipedia.org/wiki/Exact_cover) being one of [Karp's
21 NP-complete problems](https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problems), your optimization problem is NP-hard. 


