Lecture Notes Discrete Optimization - Applied Mathematics

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Figure 5: On the left: Input graph G with capacities c : E → R + . Only the edges with
positive capacity are shown. On the right: A flow f of G with flow value | f|=19. Only
the edges with positive net flow are shown.
tion constraints is the same as the total flow into t):
| f|= ∑ f(s,v).
v∈V
An example of a network and a flow is given in Figure 5.
The maximum flow reads as follows:
Maximum Flow Problem:
Given:
Goal:
A directed graph G=(V,E) with capacities c : E →R + , a source node
s∈ V and a destination node t ∈ V.
Compute an s,t-flow f of maximum value.
We introduce some more notation. Given two sets X,Y ⊆ V , define
f(X,Y)= ∑
∑
x∈X y∈Y
f(x,y).
We state a few properties. (You should try to convince yourself that these properties hold
true.)
Proposition 5.1. Let f be a flow in G. Then the following holds true:
1. For every X ⊆ V, f(X,X)=0.
2. For every X,Y ⊆ V, f(X,Y)=− f(Y,X).
3. For every X,Y,Z ⊆ V with X∩Y = /0,
f(X∪Y,Z)= f(X,Z)+ f(Y,Z) and
f(Z,X∪Y)= f(Z,X)+ f(Z,Y).
5.2 Residual Graph and Augmenting Paths
Consider an s,t-flow f . Let the residual capacity of an edge e=(u,v) ∈ E with respect
to f be defined as
r f (u,v)=c(u,v)− f(u,v).
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