given a bipartite graph $G(U,V,E\subseteq U\times V)$ with strictly positive edge-weights; is there an established name for the the task of calculating the lightest spanning subgraph and what is the best known algorithmic complexity?


Put more colloquially: if there are $p$ persons and $t$ tasks with possibly different cardinalities and a value that reflects how beneficial it is if person $p\_i$ takes over responsibility for task $t\_j$, which assignment is most beneficial under the constraint that each person must be responsible for at least task and every task must have at least one responsible person but not more than the number of tasks a person can handle or the number of persons that can collaborate on a task - these upper bounds are found in the vertex degrees.



Name for a type of assignment task
This is the [transportation problem](https://en.wikipedia.org/wiki/Transportation_theory_(mathematics)). The people correspond to supply nodes, with a lower bound of 1 and upper bound equal to the number of tasks that person can handle. The tasks correspond to demand nodes, with a lower bound of 1 and upper bound implied by the degree.


Not sure about the state-of-the-art, but the [network simplex algorithm](https://en.wikipedia.org/wiki/Network_simplex_algorithm) should be competitive.


