title: Edmonds-Karp Algorithm for a graph which has nodes with flow capacities
body: <p>I am implementing this algorithm for a directed graph. But the interesting thing about this graph nodes have also their own flow capacities. I think, this subtle change of the original problem must be handled in a special way. Because, In original max-flow problem It was okay to find any path from start to finish(actually, in Edmonds-Karp algorithm, we need to do BFS, and choose the first path that reaches the final node) But with this node-capacity extension, we need to be more careful about 'this path selection' job. I know it because, I implemented the original-algorithm and found myself getting smaller flow values than max-flow, I doubt that it has to do with this node-capacity restrictions.</p>
  
  <p>I put a lot effort on this and came up with some ideas like transforming the initial graph into a graph which has no capacity constraint on nodes by adding self loops (adding self loops  to each node and finding paths which includes this self-loops for each node on the path) or adding virtual nodes and edges whose weights supersede the initial node-capacity constraints) However, I am not convinced that any of these are nice solution for this problem.</p>
  
  <p>Any idea would be much appreciated.</p>
  
  <p>Thanks in advance. </p>