I am working on a related Steiner tree problem that I have reduced to Minimum Set Cover, but stumbled across this related problem and got stuck.
Given an universe of $n$ elements $U = \{1,2,\ldots,n\}$, let $F_1$ be a family of subsets on $U$, that covers the elements, and $msc_1$ be its minimum set cover. Let $F_2\subset F_1$ such that $F_2$ also covers the elements and $msc_2$ be its minimum set cover. Now, if it is known that every element can be in at most $f_1$ and $f_2$ sets in $F_1$ and $F_2$ respectively and every set in $F_1$ and $F_2$ have a maximal size of $k_1$ and $k_2$ respectively, then can we put a bound between $msc_2$ and $msc_1$ ?
A trivial bound is $|msc_1| \leq |msc_2| \leq F_2$ (since $F_2$ is itself a set cover of $F_1$), and $msc_2 \leq n$. But can we do better than that? Can we have something like $|msc_2| \leq h(f_1,f_2,k_1,k_2) |msc_1|$? ($h$ is a function on $f_1,f_2,k_1$ and $k_2$). Are any works in this area known? I tried to find papers dealing with multiple families of sets, but to no avail.
The problem seems so structured and since there are lots of information available on the same, I feel this can be done. But I am really struck and would be grateful for any help in this regard. Thank you.
