The Minimum Concave Cost Network Flow Problem with fixed numbers of sources and nonlinear arc costs

We prove that the Minimum Concave Cost Network Flow Problem with fixed numbers of sources and nonlinear arc costs can be solved by an algorithm requiring a number of elementary operations and a number of evaluations of the nonlinear cost functions which are both bounded by polynomials inr, n, m, wherer is the number of nodes,n is the number of arcs andm the number of sinks in the network.On leave from Institute of Mathematics, P.O. Box 631, Bo Ho, Hanoi, Vietnam.