Efficiently finding the intersections of sets that yield a desired set
Your problem is equivalent to enumerating all inclusion-minimal set covers, a problem addressed here.

Suppose that you're looking for all collections of sets whose intersection is some set $T$. All the relevant sets must contain $T$, so you can just consider all sets containing $T$, and remove $T$ from all of them. This reduces to the case $T = \emptyset$.

Now let $U$ be the union of all sets. The intersection of a collection of sets is empty iff the union of their complements is $U$. In other words, if you complement all sets, then you're interested in inclusion-minimal set covers.