Given N sets of disjoint subsets, find a set of disjoint subsets such that it satisfies a criteria
Without loss of generality, each asub$i$ can be taken to have exactly one element (size 1).  (If there is any solution, then there is a solution where all asub$i$'s are of size 1.)

Now it's easy to iterate over all sets of size 1 and determine which are valid for asub$i$.  Let $A$ be the collection of all such sets.  There is your solution.  As a bonus, I even gave you a solution where $A$ is as large as possible.

The running time is polynomial (in fact, quadratic), so this should be an efficient algorithm.