I have a weighted bipartite graph consisting of two sets $S$ and $P$. ($|S| > |P|$). I need to find a matching so that every node $s$ in $S$ matches a node of $P$. But a node $p$ in $P$ can match multiple nodes of $S$. Each $p$ has an individual limit of how many nodes it can match. The total weight of the matching should be as small as possible.
My research ended quite quick because I do not know the right key words to search for. Does this kind of matching problem have a specific name?
Second question: Do you already know some algorithms I should have a glance at?