Fastest way to perform subset test operation on a large collection of sets with same domain

Question

Assume we have trillions of sets stored somewhere. The domain for each of these sets is the same. It is also finite and discrete. So each set may be stored as a bit field (eg: 0

Answer 1

If you can preprocess the sets, the subset relation is representable as a DAG (because you're describing a poset). If the transitive reduction is computed, then I think you can avoid testing all the sets by just performing a DFS starting from the biggest sets and stopping whenever Y is no longer a subset of the current set being visited.