Suppose we have a flow network, with capacity constraints on *weighted sums* of arc flows, such as:

$$2 f(1, 2) + 3 f(4, 5) + f(3, 7) \leq 10,$$

where $f(1, 2)$ denotes the flow through arc $(1, 2)$.

**Edit**: the capacity constraints are disjoint. That is, if S is a set of pairwise disjoint subsets of arcs we have:
$\forall B \in S : \sum_{a \in B} c_a f(a) \leq C_B$

Can the problem of computing a maximum flow (or min-cost max flow) for these networks be reduced in a straightforward way to a problem where we have a capacity constraint per arc?

I've found a similar but unanswered question from 2012 here, and Google pointed me towards some articles on *shared flow*, but this problem seems to be slightly different. Also, parametric max flow seems related, but I don't see how it matches this problem.

Edit: I've just found out about polymatroidal flows, but there seems to be little introductory material. I'd be happy if someone could point me towards an introductory text.