Max flow with restriction of individual flows What is known about the complexity of solving max flow with restrictions on individual flows for some nodes? 

More precisely, in addition to wanting the max flow, you have a constraint for some vertices $S\subset V$ where $\forall s \in S, \forall u\in \{u | (u,s) \in E \} , f(u,s) \leq \frac{1}{2} \sum_{u} f(u,s) $

Meaning no one edge contributes more than half the total flow into a node. 

Also, the flows must be integral [and the capacities are integral]
That constraint is linear, so the entire problem is an instance of linear programming, thus can be solved in polynomial time.  (I am assuming there is no restriction or requirement for integer flows.)

In general if you allow arbitrary constraints, the problem can become NP-hard.