Algorithms for Solving the Independent. Flow Problems

Given a capacitated network with the entrance· vertex set VI and the exit-vertex set V2 on which polymatroids are defmed, an independent flow is a flow in the network such that a vector corresponding to the supplies in VI and a vector corresponding to the demands in V2 are, respectively, independent vectors of the polymatroids on VI and V2 • The independent-flow problems considered in the present paper are the following two: (1) to find a maximum independent flow; and (2) to find an optimal independent flow, i.e., a maximum independent flow of the minimum cost when a cost is given to each arc. We present several theorems which algorithmically characterize optimal independent flows and we propose algorithms for solving the independent-flow problems based on the theorems.