I have a set $S$ and a set $P = \{P_{1},...,P_{n}\}$ with $\bigcup P_{i} = S$. I want to find all the inclusion-minimal subsets of $P$ that are covers of $S$.
What is the best algorithm for enumerating all the inclusion-minimal covers of $S$ contained in a set $P$ and what is its running time?
Additional information : The algorithm must work in the general case. $|S|$ and $n$ can be anything. I need to enumerate the exact set of all the inclusion-minimal covers of $S$ contained in $P$. Obviously, it is possible to find the first one in polynomial time by just removing elements of $P$. The second one could be found easily by using the same method starting with the different $P\setminus \{P_{i}\}$ with $P_{i}$ an element of the first cover.
I don't really hope for a polynomial delay algorithm but I would be really happy with an incremental polynomial delay, if such a thing exists.