Balancing Applied to Maximum Network Flow

We explore balancing as a definitional and algorithmic tool for finding minimum cuts and maximum flows in ordinary and parametric networks. We show that a standard monotonic parametric maximum flow problem can be formulated as a problem of computing a particular maximum flow that is balanced in an appropriate sense. We present a divide-and-conquer algorithm to compute such a balanced flow in a logarithmic number of ordinary maximum-flow computations. For the special case of a bipartite network, we present two simple, local algorithms for computing a balanced flow. The local balancing idea becomes even simpler when applied to the ordinary maximum flow problem. For this problem, we present a round-robin arc-balancing algorithm that computes a maximum flow on an n-vertex, m-arc network with integer arc capacities of at most U in O(nm log(nU)) time. Although this algorithm is slower by at least a factor of n than other known algorithms, it is extremely simple and well-suited to parallel and distributed implementation.