An Extension of a Balanced Flow in a Network

We say x is an element of a network N, if x is an edge or a vertex of N. We consider a flow network N in which capacities and distribution parameters are given on the elements in N. A d-balanced flow is defined as a flow such that the flow value at each element x is not more than d(x)⋅f, where f is the total amount of the flow and d(x) is the given distribution parameter of x. We give a max-flow min-cut theorem for this d-balanced flow. Then we show a method for evaluating the maximum value of d-balanced flow between a pair of vertices. The algorithm gives the maximum value of d-balanced flow s to t in N with at most 2|X| evaluations of maximum flow in a network, where X is the element set of N.