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2

If you are willing to relax the 2 to some other constant, then yes: this can be solved in $O(c^a)$ time for some absolute constant $c$. You can view this problem as isomorphism of hypergraphs, where the vertices in $A_i$ are the vertices of your hypergraph, and the vertices in $B_i$ are hyperedges. Luks (STOC '99) showed how to solve hypergraph isomorphism ...


3

Something like this is used in the definition of creative sets by Joseph and Young (1985). See the Wikipedia article on polynomial creativity. But their use is to concoct NP-hard sets with unusual properties (potential counterexamples to the Berman–Hartmanis conjecture rather than to say anything nontrivial about the complexity of any natural algorithmic ...


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