An efficient method to compute a minimum set of sets that form the union of these sets? Let's say we have a set of sets:

$$\mathfrak S = \lbrace S_1, S_2, ... , S_n\rbrace$$

And the union of the all the sets in this set:

$$\mathfrak U = \bigcup\limits_{i=1}^{n} S_{i}$$

And so there is at least one minimum set of sets that is identical to this union:

$$(\mathfrak M \subseteq  \mathfrak S) = \mathfrak U$$

What is the most efficient known method of computing $\mathfrak M$ ?
In case $S_i$ sets are finite your problem is reduced to the Set Cover problem which is NP-complete. You can also attack this problem by any approximation algorithm, e.g. Greedy algorithm, or by using Linear programming.