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We consider a compound problem of the generalized minimum-cost flow problem and the independent-flow problem, which we call the generalized independent-flow problem. The generalized minimum- cost flow problem is to find a minimum-cost flow in a capacitated network with gains, where each arc flow is multiplied by a gain factor when going through an arc. On the other hand, the independent-flow problem due to Fujishige is to find a minimum-cost flow in a multiple-source multiple-sink capacitated network with submodular constraints on the set of supply vectors on the source vertex set and on the set of demand vectors on the sink vertex set. We present a polynomial-time algorithm for the generalized independent- flow problem, based on Wayne's algorithm for generalized minimum-cost flows and Fujishige's algorithm for independent flows, which can be regarded as an extension of Wallacher and Zimmermann's submodular flow algorithm.

A POLYNOMIAL-TIME ALGORITHM FOR THE GENERALIZED INDEPENDENT-FLOW PROBLEM