optimal flow for index multiple source and destination in directed graph I am facing the similar problem to max flow in multiple source-destination directed graph (which has a familiar solution of connecting all the sources to one source and the same for the destination, and solving it with some algorithms like Ford-Fulkerson), but the difference in my problem is that :


all edges has capacity of 1 (the flow is discrete so once an edge is used - it can't be used again)
each source need to flow to a specific destination (e.g. $v_1$ needs to flow to sink $u_1$ - other sinks can't consume $v_1$ flow).


I need to find the max sources that can reach to their destination on a given graph. Is there a solution for this or I have to compute all the possiblities ?
This is an instance of a multi-commodity flow problem.  Unfortunately the problem is NP-hard (even when all edge capacities are 1).