Max-Balancing Weighted Directed Graphs and Matrix Scaling

A weighted directed graph G is a triple V, A, g where V, A is a directed graph and g is an arbitrary real-valued function defined on the arc set A. Let G be a strongly-connected, simple weighted directed graph. We say that G is max-balanced if for every nontrivial subset of the vertices W, the maximum weight over arcs leaving W equals the maximum weight over arcs entering W. We show that there exists a up to an additive constant unique potential pv for v ∈ V such that V, A, gp is max-balanced where gap = pu + ga-pv for a = u, v ∈ A. We describe an O|V|2|A| algorithm for computing p using an algorithm for computing the maximum cycle-mean of G. Finally, we apply our principal result to the similarity scaling of nonnegative matrices.