IE170: Algorithms in Systems Engineering: Lecture 25
Jeff Linderoth
Department of Industrial and Systems Engineering
Lehigh University
March 30, 2007
Jeff Linderoth
(Lehigh University)
IE170:Lecture 25
Lecture Notes
1 / 23
Taking Stock
Last Time
Flows
This Time
(Cardinality) Matching
Homework and Review
Jeff Linderoth
(Lehigh University)
IE170:Lecture 25
Lecture Notes
2 / 23
Flows
The Big Kahuna
MaxFlow MinCut Theorem
The following statements are equivalent
1
f
is a maximum flow
2
f
admits no augmenting path. (No
(
s, t
)
path in residual network)
3

f

=
c
(
S, T
)
for some cut
(
S, T
)
Jeff Linderoth
(Lehigh University)
IE170:Lecture 25
Lecture Notes
3 / 23
Flows
FordFulkerson Algorithm
This gave Lester Ford and Del Fulkerson an idea to find he maximum
flow in a network:
FordFulkerson
(
V, E, c, s, t
)
1
for
i
←
1
to
n
2
do
f
[
u, v
]
←
f
[
v, u
]
←
0
3
while
∃
augmenting path
P
in
G
f
4
do
augment
f
by
c
f
(
P
)
Assume all capacities are integers. If they are rational numbers, scale
them to be integers.
Jeff Linderoth
(Lehigh University)
IE170:Lecture 25
Lecture Notes
4 / 23
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Flows
Analysis
If the maximum flow is

f

*
, then (since the augmenting path must
raise the flow by at least 1 on each iteration), we will require
≤ 
f

*
iterations.
Augmenting the flow takes
O
(

E

)
FordFulkerson
runs in
O
(

f

*

E

)
This is
not
polynomial in the size of the input.
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 Spring '07
 Ralphs
 Systems Engineering, Lehigh University, Shortest path problem, Flow network, Bipartite graph, Maxflow mincut theorem, Jeff Linderoth

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