Popular Matchings with Multiple Partners

Our input is a bipartite graph $G = (R \cup H,E)$ where each vertex in $R \cup H$ has a preference list strictly ranking its neighbors. The vertices in $R$ (similarly, in $H$) are called residents (resp., hospitals): each resident seeks to be matched to a hospital while each hospital $h$ seeks $\mathsf{cap}(h) \ge 1$ many residents to be matched to it. The Gale-Shapley algorithm computes a stable matching in $G$ in linear time. We consider the problem of computing a popular matching in $G$ - a matching $M$ is popular if $M$ cannot lose an election to any other matching where vertices cast votes for one matching versus another. Our main contribution is to show that a max-size popular matching in $G$ can be computed by the 2-level Gale-Shapley algorithm in linear time. This is a simple extension of the classical Gale-Shapley algorithm and we prove its correctness via linear programming.