Balanced Network Flows

Let G be a simple, undirected graph. A special network N, called a balanced network, is constructed from G such that maximum matchings and f-factors in G correspond to maximum flows in N. A max-balancedflow-min-balanced-cut theorem is proved for balanced networks. It is shown that Tutte’s Factor Theorem is equivalent to this network flow theorem, and that f-barriers are equivalent to minimum balanced edgecuts. A max-balanced-flow algorithm will solve the factor problem. 1. Balanced Networks. Let G be a simple graph, directed or undirected, with vertex set V(G) and edge set E(G). A network N is a directed graph which contains two special vertices s and t, the source and target, respectively, and in which every edge e is assigned a positive integervalued capacity cap(e). Terminology for graphs, networks, and flows is taken from Bondy and Murty [1]. Edges of a directed graph are ordered pairs of vertices. If (u,v) is an edge, we indicate that u is adjacent to v by u→v. Edges of an undirected graph are unordered pairs, and we write the pair {u,v} as uv, where the order is unimportant. In a directed graph we also often use uv to indicate one of the edges (u,v) or (v,u), especially if the direction is not explicitly given. The opposite direction will then be given by vu. Let f be an integervalued function on E(N). Given any u∈V(N), the out-flow at u is f+(u) = ∑ v, u→v f(uv) and the in-flow at u is f–(u) = ∑ v, v→u f(vu). The function f is called a flow if it satisfies the two conditions: i) f+(u) = f–(u), for all vertices u≠ s,t, (the conservation condition) ii) 0 ≤ f(uv) ≤ cap(uv), for all edges uv (the capacity constraint). The value of the flow is val(f) = f+(s)–f–(s), that is, the net out-flow at the source s. We are interested in a special kind of network, called a balanced network. * This work was supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada.