Min-cost flow with per-edge flow conservation

Question

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

Answer 1

There is a well-studied problem close to your one: Integral Flow With Multipliers. It was proved to be NP-hard in the seminal Sartaj Sahni's paper in computational complexity theory (see section 2.2 of the paper). Another interesting, more recent paper can be found here.