Reasoning about optimal stable matchings under partial information

We study two-sided matching markets in which participants are initially endowed with partial preference orderings, lacking precise information about their true, strictly ordered list of preferences. We wish to reason about matchings that are stable with respect to agents' true preferences, and which are furthermore optimal for one given side of the market. We present three main results. First, one can decide in polynomial time whether there exists a matching that is stable and optimal under all strict preference orders that refine the given partial orders, and can construct this matching in polynomial time if it does exist. We show, however, that deciding whether a given pair of agents are matched in all or no such optimal stable matchings is co-NP-complete, even under quite severe restrictions on preferences. Finally, we describe a polynomial-time algorithm that decides, given a matching that is stable under the partial preference orderings, whether that matching is stable and optimal for one side of the market under some refinement of the partial orders.