  We study the classical problem of matching $n$ agents to $n$ objects, where
the agents have ranked preferences over the objects. We focus on two popular
desiderata from the matching literature: Pareto optimality and rank-maximality.
Instead of asking the agents to report their complete preferences, our goal is
to learn a desirable matching from partial preferences, specifically a matching
that is necessarily Pareto optimal (NPO) or necessarily rank-maximal (NRM)
under any completion of the partial preferences. We focus on the top-$k$ model
in which agents reveal a prefix of their preference rankings. We design
efficient algorithms to check if a given matching is NPO or NRM, and to check
whether such a matching exists given top-$k$ partial preferences. We also study
online algorithms for eliciting partial preferences adaptively, and prove
bounds on their competitive ratio.
