Popular Matchings with One-Sided Bias

Let $G = (A \cup B,E)$ be a bipartite graph where the set $A$ consists of agents or main players and the set $B$ consists of jobs or secondary players. Every vertex has a strict ranking of its neighbors. A matching $M$ is popular if for any matching $N$, the number of vertices that prefer $M$ to $N$ is at least the number that prefer $N$ to $M$. Popular matchings always exist in $G$ since every stable matching is popular. A matching $M$ is $A$-popular if for any matching $N$, the number of agents (i.e., vertices in $A$) that prefer $M$ to $N$ is at least the number of agents that prefer $N$ to $M$. Unlike popular matchings, $A$-popular matchings need not exist in a given instance $G$ and there is a simple linear time algorithm to decide if $G$ admits an $A$-popular matching and compute one, if so. We consider the problem of deciding if $G$ admits a matching that is both popular and $A$-popular and finding one, if so. We call such matchings fully popular. A fully popular matching is useful when $A$ is the more important side -- so along with overall popularity, we would like to maintain ``popularity within the set $A$''. A fully popular matching is not necessarily a min-size/max-size popular matching and all known polynomial-time algorithms for popular matching problems compute either min-size or max-size popular matchings. Here we show a linear time algorithm for the fully popular matching problem, thus our result shows a new tractable subclass of popular matchings.