1
Contents
Maximum flow problem.
Minimum cut problem.
Maxflow mincut theorem.
Augmenting path algorithm.
Shortest augmenting path.
Capacityscaling.
Augmenting Path
Augmenting path = path in residual graph.
Max flow
⇔
no augmenting paths ???
s
4
2
5
3
t
10
13
10
4
4
4
4
G
s
4
2
5
3
t
10
10
10
4
4
4
4
3
G
f
10
10
10
Flow value = 14
MaxFlow MinCut Theorem
Augmenting path theorem (FordFulkerson, 1956):
A flow
f
is a max flow if and only if there are no augmenting paths.
MAXFLOW MINCUT THEOREM (FordFulkerson,
1956):
the value of the max flow is equal to the value of the
min cut.
We prove both theorems simultaneously by showing the
TFAE:
(i)
f is a max flow.
(ii)
There is no augmenting path relative to f.
(iii)
There exists a cut (S, T) such that 
f
 = c(S, T).
Proof
We prove both simultaneously by showing the TFAE:
(i)
f is a max flow.
(ii)
There is no augmenting path relative to f.
(iii)
There exists a cut (S, T) such that f = c(S, T).
(i)
⇒
(ii)
Let f be a flow. If there exists an augmenting path, then we can improve f by
sending flow along path. (proof by contradiction)
(iii)
⇒
(i)
This was the Corollary to Lemma 2.
Proof
(ii)
⇒
(iii)
Let
f
be a flow with no augmenting paths.
Let S be set of vertices reachable from s in residual graph.
Let T=V
−
S. Is t
∈
T. ?
(clearly s
∈
S, and t
∉
S by definition of
f
, o.w.
∃
augmenting path)
For each u
∈
S, v
∈
T,
f
(u,v) = c(u,v). ?
(o.w. c
f
(u,v)=c(u,v)
−
f
(u,v)>0, then v
∈
S).
Therefore: ?

f
 =
f
(S,T) = c(S,T).
Residual Network
s
t
S
T
FordFulkerson algorithm
Let G=(V,E) be a flow network.
FordFulkerson
(G,s,t,c)
01
for
each edge (u,v) in E
do
02
f(u,v)
←
f(v,u)
←
0
03
while
there exists a path p from s to t in residual
network G
f
do
04
c
f
= min{c
f
(u,v): (u,v) is in p}
05
for
each edge (u,v) in p
do
06
f(u,v)
←
f(u,v) + c
f
07
f(v,u)
←
f(u,v)
08
return
f
Cost?
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 Spring '09
 Flow network, Maximum flow problem, Maxflow mincut theorem, shortest augmenting path

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