The successive shortest path algorithm, used to solve the minimum-cost flow problem, can be described as follows :
Successive shortest path (for minimum-cost flow) :
while all flow is not sent :
compute shortest path between source and sink nodes w.r.t. arc costs (through Dijkstra's algorithm for example).
augment the most flow possible through this path (and add reverse arcs on this path).
compute reduced costs on arcs in the network (through updating node potentials).
Is computing reduced costs necessary, i.e., does the algorithm function correctly without modifying any arc costs?
Edit : The algorithm gives the same solution, with or without reduced costs computation, on some example networks done by hand.