Greedy concepts for network flow problems

A directed graph G = ( V , E ) is given by a finite vertex set V and an edge set E C V× V. We choose two distinct vertices s e Vand t~ Vthat we consider as source and sink respectively. A path (from s to t) is a sequence of distinct edges (s, xl ), (xl, Xz) . . . . . (Xrj, Xr), (Xr, t) for some r. Denote by P the set of all such paths. For an edge e e E let PeCP be the set of paths passing through this edge. With every edge e ~ E we associate a nonnegative integer capacity b e and an arbitrary positive cost c e. For a path p e P the capacity and cost is defined by bp = min be, (1) e ~ p