1. Feasible Flow Network : A digraph with a nonnegative capacity c(e) on each edge e and a distinguished source node s and sink node t. Flow: A flow f assign a value f(e) to each edge e. f + (v) denotes the total flow on edges leaving v and f - (v) denotes the total flow on edges entering v. Feasible Flow: A flow is feasible if it satisfies the capacity constraints 0<=f(e)<=c(e) for each edge and the conservation contraints f + (v)= f - (v) for each node v  {s,t}. Maximum flow: The value vaf(f) of a flow f is the net flow f - (t)-f + (t) into the sink. A maximum flow is a feasible flow of maximum value.

2. Example of Network Flow

3. f-augmenting path f-augmenting path : When f is a feasible flow in a network N, an f-augmenting path is a source-to-sink path P in the underlying graph G such that for each e  E(P), (a) if P follows e in the forward direction, then f(e)<c(e). (b) if P follows e in the backward direction, then f(e)>0. Tolerance of P: Let  (e)=c(e)-f(e) when e is forward on P, and let  (e)=f(e) when e is backward on P. The tolerance of P is min e  E(P)  (e).

4. Lemma 4.3.5 If P is an f-augmenting path with tolerance z, then changing +z on edges followed forward by P and by – z on edges followed backward by P produces a feasible flow f’ with val(f’)=val(f)+z.

5. Source/Sink Cut [S,T] Source/Sink Cut [S,T]: In a network, a source/sink cut[S,T] consists of the edges with tail in a source set S and head in a sink set T, where S and T partition the set of nodes, with s  S and t  T. cap(S,T): The capacity of the cut[S,T], written cap(S,T), is the total of the capacities on the edges of [S,T].

6. Lemma 4.3.7 If f is a feasible flow and [S,T] is a source/sink cut, then the net flow out of S and net flow into T equal val(f). Proof. 1. f + (S)-f - (S)=  v  S [f + (v)-f - (v)]= f + (s)-f - (s)=val(f). 2. f + (T)-f - (T)=  v  T [f + (v)-f - (v)]= f + (t)-f - (t)=-val(f).

7. Corollary 4.3.8 If f is feasible flow and [S,T] is a source/sink cut, then val(f)<=cap(S,T). Proof. 1. f + (S)-f - (S)<=cap(S,T) due to the capacity constraints.  val(f)<=cap(S,T) by Lemma 4.3.7.

8. Ford-Fulkerson Labeling Algorithm

9. Example 4.3.10

10. Theorem 4.3.11 Ford-Fulkerson Labeling Algorithm finds a maximum flow and a minimum source/sink cut. Proof. 1. We need to prove val(f)=cap(S,T) when the algorithm terminates. 2. When the algorithm terminates: 2-1. s  S and t  R=S. 2-2. No edge from S to T has excess capability, and no edge from T to S has nonzero flow in f.  val(f)=f + (S)-f - (S)=cap(S,T).