I am trying to solve a linear program that is identical to a min-cost flow problem, except for a difference in the flow-conservation constraint.
Instead of the summed outgoing flow equaling the summed incoming flow for each node:
$$\sum_i f_{i,n} = \sum_j f_{n, j} \ \forall n$$
The flow of each outgoing edge should equal the summed incoming flow for each node:
$$\sum_i f_{i,n} = f_{n, m} \ \forall n, m$$
In other words: The incoming flow is replicated along all outgoing edges.
I wonder if:
1.) This problem has a particular name in the literature
2.) It has an optimal integral solution when all edge capacities are integral