When solving a multi source/sink max flow problem, the classic reduction to the single source/sink problem is to add a new source and sink node with infinite capacities that connect to/from the existing source/sink nodes.
I was curious about an alternate reduction. It is kind of silly buy I just wanted to sanity check it.
Is it possible to group all the source nodes into one big source node, with all the outgoing edges of the original source nodes now added to the mega source node. We can collapse repeated edges by adding up their capacities. A similar thing can be done for sink nodes.
Let's say I have a solution to the mega source/sink problem. You would have to keep track of which edges in the mega source/sink node correspond to which edges in the original sources/sinks, and choose the same corresponding flow for the original source nodes' edges. For two edges that you collapsed in the mega node by summing their capacities, you can just distribute the flow arbitrarily amongst the original source nodes (maintaining the capacity constraints).
Does this approach fail catastrophically or sound reasonable?