Ford-Fulkerson Max Flow Labeling Algorithm

The Ford-Fulkerson max ow labeling algorithm[3, 4] was introduced in the mid-1950's, and became the seminal work that is still applicable. The material presented in this note is taken from their book[5]. We are given a simple network with two speci ed nodes: source (s) and sink (t). Since the network is assumed to be simple, an arc is identi ed by its endpoints: (x; y) is the arc from node x to node y. A ow across arc (x; y) is denoted by f(x; y), and the arc's capacity is c(x; y). The ow must satisfy 0 f(x; y) c(x; y). The algorithm has two parts, which Ford and Fulkerson called Routine A and Routine B, respectively. The rst is a labeling process that searches for a ow augmenting path { i.e., a path from s to t for which f < c along all forward arcs and f > 0 along all backward arcs. If Routine A nds a ow augmenting path, Routine B changes the ow accordingly. Otherwise, no augmenting path exists, and optimality of the current ow is ensured by their theorem: