Matching on bipartite graph - multiple edges I have a weighted bipartite graph consisting of two sets $S$ and $P$. ($|S| &gt; |P|$).
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?
According to the comments above, the problem can be reduced to a weighted bipartite matching / assignment problem.

If $f(p)$ specifies the limit of matches for a node in $P$, we replace each $p$ by $f(p)$ nodes.
To assure $|S| = |P|$ we can add dummy nodes to $S$.

Now we can solve the problem by a standard algorithm. E.g. Hungarian Method.