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