Total flow using minimum number of edges on a bipartite network If I have a set of sources $S$ with total capacity $C$ and a set of sinks $T$ with the same capacity $C$, but not necessarily the same cardinality, is there an efficient way to find the minimum number of direct edges between sources and sinks and their required capacity, assuming $flow=C$?

TIA, Luis
Your problem is NP-hard.
Consider a partition problem instance with input $a_1,\ldots,a_n$.
We create a complete bipartite graph with $2$ source vertices each with capacity $\frac{1}{2} \sum_{i=1}^n a_i$, and $n$ sink vertices, where the $i$th vertex has capacity $a_i$. Each edge has infinite capacity. 

It's easy to verify there is a partition with equal sum if and only if the solution to your problem on $G$ has $n$ edges.