ON AN EXTENSION OF THE MAXIMUM-FLOW MINIMUM-CUT THEOREM TO MULTICOMMO))ITY FLOWS

The extension of the maximum-flow minimum-cut theorem for singlecommodity network flows to the case of multicommodity flows has been discussed by many researchers (see III for the details). RecQntly, K. Onaga established a necessary and sufficient condition for the existence of a feasible multicommodity flow configuration on a capacity-constrained undirected network when the locations of source and sink as well as the total flow value are given for each commodity [2]. His condition seems to be of fundamental significance in multicommodity network-flow theory although it is not combined with a practical computational algorithm. However, the proof developed in [2] is purely graphical and fairly complicated. In the present paper, a simple proof to his condition will be given based on the duality theorem in linear programming, where some improvement will be made also on the statement of the condition itself.