An Algorithm for Balanced Flows

The Balanced Network Search (BNS) is an algorithm which finds a maximum balanced flow in a balanced network N. This algorithm is a way of using network flows to solve a number of standard problems, including maximum matchings, the factor problem, maximum capacitated b-matchings, etc., in general graphs. The value of a maximum balanced flow equals the capacity of a minimum balanced edge-cut. Flow-carrying balanced networks contain structures called generalized blossoms. They are not based on odd cycles. Rather they are the connected components of a residual sub-network of N. An algorithm is given for finding a maximum balanced flow, by constructing complementary pairs of valid augmenting paths. 1. Balanced Networks. Balanced networks were introduced in [7]. They are special bipartite directed graphs. Let X={x1, x2, ... xn} and Y={y1, y2, ... yn} be two sets of vertices. Then a balanced network N with vertices X∪Y has two additional vertices, the source s and the target t. A directed edge is a pair (u,v). Its capacity is a non-negative integer, denoted cap(uv). There is an edge (s,xi) for each xi∈X and an edge (yi,t) for each yi∈Y. All other edges (ui,vj ) have one end in X and one end in Y, so either ui∈X, vj∈Y, or ui∈Y, vj∈X. Fig. 1 shows an example of a balanced network (although the capacities are not shown in the diagram). The vertices of N can be divided into complementary pairs. Vertices s and t are complementary ; so are xi and yi. Write s′=t, t′=s, xi′ = yi, and yi′ = xi to indicate complementarity. The edges can also be divided into complementary pairs. Edges (s,xi) and (yi,t) are complementary edges. So are (xi,yj ) and (xj ,yi). In general, (u,v)′=(v ′,u′). Complementary edges must always have equal capacities . Thus the network N is said to be balanced .