In the set cover (SC) problem we are given a universe $U$ (a set) of $n$ elements and a collection $S$ of $m$ sets whose union equals the universe. The set cover problem is to identify the smallest sub-collection of $S$ whose union equals the universe.
If I add the constraint that $m=n$ to SC, is this new problem still NP-hard?
My attempt is as follows:
To construct an instance of the new problem we do:
- If $n=m$, then we are done.
- If $n>m$, then we simply add $n-m$ dummy sets to the collection $S$ and we are done.
- If $n<m$, then ... (I am stuck). But I think since the union of the $m$ sets is $U$ then we must have that $S$ contains redundant sets. Is this OK?