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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.
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Example of Network Flow
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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).
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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.
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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].
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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).
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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.
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Ford-Fulkerson Labeling Algorithm
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Example 4.3.10
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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).
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