An $O(n^3)$ time algorithm for the maximum-weight limited-capacity many-to-many matching

Given an undirected bipartite graph $G=(A \cup B, E)$, a many-to-many matching (MM) in $G$ matches each vertex $v$ in $A$ (resp. $B$) to at least one vertex in $B$ (resp. $A$). In this paper, we consider the limited-capacity many-to-many matching (LCMM) in $G$, where each vertex $v\in A\cup B$ is matched to at least one and at most $Cap(v)$ vertices; the function $Cap : A\cup B \rightarrow \mathbb{Z}> 0$ denotes the capacity of $v$ (an upper bound on its degree in the LCMM). We give an $O(n^3)$ time algorithm for finding a maximum (respectively minimum) weight LCMM in $G$ with non-positive real (respectively non-negative real) edge weights, where $\lvert A \rvert+\lvert B \rvert=n$.