Fig 9. An input flow network with a subpath constraint {2-4-7}.
(a), the number next to an edge is the edge cost. For every edge e, the edge constraint implies 1 ≤ f(e) ≤ inf. (b), the transformed min-flow circulation network. The 2-tuple (a, b) next to each edge indicates the optimal flow on the edge and the edge cost respectively. After Step 3, the path constraints set is Psub = {(1, 2), (1, 3), (2, 4, 7), (4, 5), (4, 6), (5, 8), (6, 8), (7, 8)}. Two edges no longer in the constraint set are shown in green. For these two edges, the minimum flow requirement is 0; for the rest of edges, it is 1. Two dummy nodes, s and t, are added to complete the circulation. The number of flows after decomposition is equal to the minimum flow which is 3.