# Is finding a set cover of size $k$ NP-complete?

I know that set cover problem is NP-complete. We are given a universe $U$ (a set) of $n$ elements and a collection $S$ of $m$ sets whose union equals the universe and an integer $k$. The set cover problem is to identify a sub-collection of $S$ of size $k$ or less whose union equals the universe?

Now if I remove or less, and ask instead that is there a sub-collection of $S$ of size $k$ whose union equals the universe? Is this still NP-complete?

Here, of course, $k<m$.

My answer is that the problem is still NP-complete. From the initial problem we simply keep the same instance and we find a set cover of size $s\leqslant k$ then the modified problem has a set cover of size $k$ if $s=k$.

Any help?

• For any instance, there is a Set Cover of size $k$ if and only if there is a set cover of size at most $k$. (Because, given any set cover of size less than $k$, you can add arbitrary sets to it to obtain a set cover of size $k$.) So, the "or less" makes no difference at all - the problem is exactly the same with or without it. The special-purpose reductions in the answer below are not necessary. – Neal Young Aug 27 '18 at 12:40

Yes, it remains NP-complete, even under the restriction that all subsets of a specific size $s$ (which translates to immediately requiring that $k = n / s$.

For example, for $k = n / 3$, 3-dimensional matching can be directly reduced to your problem.

• @det YW, cool you ended up finding it yourself :). – aelguindy Nov 6 '16 at 16:39

I have found an answer for my question in [1]. The problem I am interested in, called in [1] MINIMUM COVER, is NP-complete by reduction from 3-EXACT COVER.

The 3-EXACT COVER is defined as [1]: Given a family $F=\{S_1,\ldots,S_n\}$ of $n$ subsets of $S=\{u_1,\ldots,u_{3m}\}$, each of cardinality three, is there a subfamily of $m$ subsets that covers $S$?

The 3-EXACT COVER is NP-complete [1].

The MINIMUM COVER is defined as [1]: Given a family $F=\{S_1,\ldots,S_n\}$ of subsets of a finite set $U$, and an integer $k\leqslant n$, is there a subfamily $C$ of $F$ containing $k$ sets such that $\bigcup\limits_{S_j\in C}S_j=U$?

As stated in [1], the 3-EXACT COVER is just the special case of MINIMUM COVER in which $|S_j|=3$ for $j=1,\ldots,n$, and $k=\frac{1}{3}|U|$. Consequently, any instance of 3-EXACT COVER can be trivially transformed into an instance of MINIMUM COVER. ...

[1] C. H. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Algorithms and Complexity. Dover Publications, 1998. First published in 1982 by Prentice-Hall.