I am given a set $U = \{1, \cdots, n\}$ and a set of subsets $C_1, \cdots, C_k$ of $U$; here the sets are all the same size.
The minimum set-cover problem of course is to find the minimum number of the $C_i$ such that their union is $U$.
I'm looking at the "opposite" problem: I want to find the minimum-size subset $K$ of $U$ such that every $k \in K$ has that $k \in C_i$ for some $i$.
Edit: I researched more and this is the Minimum Satisfiability problem. One can formulate the problem as a CNF SAT formula: the formula is the conjunction of $(c_{i,1} \vee \cdots \vee c_{i,m} )$ where the $c_{i,j} \in C_i$ for all $i$.
The MinSat problem is NP-complete in the general case, but here all of the literals are positive. Are there faster algorithms for this case?