Popular b-matchings

Abstract

Suppose that each member of a set of agents has a preference list of a subset of houses, possibly involving ties and each agent and house has their capacity denoting the maximum number of correspondingly agents/houses that can be matched to him/her/it. We want to find a matching $M$, for which there is no other matching $M'$ such that more agents prefer $M'$ to $M$ than $M$ to $M'$. (What it means that an agent prefers one matching to the other is explained in the paper.) Popular matchings have been studied quite extensively, especially in the one-to-one setting. We provide a characterization of popular b-matchings for two defintions of popularity, show some $NP$-hardness results and for certain versions describe polynomial algorithms.