# Combined arc capacity constraints in network flows

Consider the network flow polyhedron for a directed graph $$G = (N, A)$$. Along with the standard flow-balance and single-arc capacity constraints, we are faced with additional constraints which enforce capacities on the total flow through subsets of arcs. Formally, for (not necessarily disjoint) subsets $$S_1, S_2, \dots, S_k \subseteq A$$ we have constraints $$\sum_{a \in S_i} x_a \leq u_{S_i} \quad \forall i \in [k].$$

Do such constraints have a specific name in the network flow literature, or have they been specifically studied? I'm mostly interested in the polyhedral aspects of these constraints, but applications would be interesting as well.