Ranking Sets of Objects: The Complexity of Avoiding Impossibility Results

The problem of lifting a preference order on set objects to family subsets this is fundamental with wide variety applications in AI. process often guided by axioms postulating properties the lifted should have. Well-known impossibility results Kannai and Peleg Barbera Pattanaik tell us that some desirable – namely dominance (strict) independence are not jointly satisfiable for any linear if all non-empty sets be ordered. On other hand, ordered, orders families sets. Such very important as they allow use orders, example, combinatorial voting. In paper, we determine computational complexity recognizing such families. We show it \Pi_2^p-complete decide given whether or strict needs total. Furthermore, remains coNP-complete can incomplete. Additionally, these problems increase exponentially explicitly but via succinct domain restriction. Finally, NP-complete at least one objects.