Introduction to Network Flow Problems

1 Basic definitions and properties Definition 1.1. A flow network is a directed graph D = (V,E) with two distinguished vertices s and t called the source and the sink, respectively. Moreover, each arc (u, v) ∈ E has a certain capacity c(u, v) ≥ 0 assigned to it. If (u, v) / ∈ E (including pairs of the form (u, u)), we assume c(u, v) = 0. In this note, we shall restrict ourselves to the case where capacities are all rational numbers. Some of the capacities might be∞. Definition 1.2. Let G = (V,E) be a graph or a digraph. Let X be a proper non-empty subset of V . Let X̄ := V −X , then the pair (X, X̄) forms a partition of V , called a cut of G. The set of edges of G with one end point in each of X and X̄ is called an edge cut of G, denoted by [X, X̄]. Definition 1.3. A source/sink cut of a network D is a cut (S, T ) with s ∈ S and t ∈ T . (Note that, implicitly T = S̄.) Definition 1.4. A flow for a network D = (V,E) is a function f : V × V → R, which assigns a real number to each pair (u, v) of vertices. A flow f is called a feasible flow if it satisfies the following conditions: (i) 0 ≤ f(u, v) ≤ c(u, v),∀(u, v) ∈ E. These are the capacity constraints. (If a capacity is∞, then there is no upper bound on the flow value on that edge.) (ii) For all v ∈ V − {s, t}, the total flow into v is the same as the total flow out of v: ∑