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undirected graph
A partition of the vertices of a graph into two
disjoint subsets
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undirected graph
A partition of the vertices of a graph into two
disjoint subsets
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undirected graph
A partition of the vertices of a graph into two
disjoint subsets
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A partition of the vertices of a graph into two
disjoint subsets
undirected graph
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Cut-set of the cut is the set of edges whose
end points are in different subsets.
undirected graph
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Cut-set of the cut is the set of edges whose
end points are in different subsets.
Cut-set
undirected graph
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weight = number of edges or sum of weight
on edges
weight is 7
undirected graph
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cut (directed)
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directed graph
A partition of the vertices of a graph into two
disjoint subsets
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9
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directed graph
A partition of the vertices of a graph into two
disjoint subsets
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9
2
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directed graph
A partition of the vertices of a graph into two
disjoint subsets
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9
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directed graph
A partition of the vertices of a graph into two
disjoint subsets
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directed graph
Cut-set of the cut is the set of edges whose
end points are in different subsets.
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directed graph
Cut-set of the cut is the set of edges whose
end points are in different subsets.
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Cut-set
directed graph
Cut-set of the cut is the set of edges whose
end points are in different subsets.
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weight is 5⇢ or 2⇠
directed graph
weight = number of edges or sum of weight
on edges
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s-t cut
1. one side is source
2. another side is sink
3. cut-set only consists of edges going
from source’s side to sink’s side
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flow network
Source Sink Other
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flow network
Source Sink
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flow network
Source Sink
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flow network
cut-set only consists of edges going
from source’s side to sink’s side
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weight is 6
flow network
cut-set only consists of edges going
from source’s side to sink’s side
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Max-Flow Min-Cut Theorem
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Observation 1
The network flow sent across any cut is equal to the
amount reaching sink.
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3/3
3/8
4/4
2/2
1/4
4/4
2/3
total flow = 6, flow on cut = 3 + 3 = 6
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Observation 1
The network flow sent across any cut is equal to the
amount reaching sink.
1
3
2 4
5
6
3/3
3/8
4/4
2/2
1/4
4/4
2/3
total flow = 6, flow on cut = 3 + 3 = 6
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Observation 1
The network flow sent across any cut is equal to the
amount reaching sink.
1
3
2 4
5
6
3/3
3/8
4/4
2/2
1/4
4/4
2/3
total flow = 6, flow on cut = 3 + 4 - 1 = 6
Slide 30
Slide 30 text
Observation 1
The network flow sent across any cut is equal to the
amount reaching sink.
1
3
2 4
5
6
3/3
3/8
4/4
2/2
1/4
4/4
2/3
total flow = 6, flow on cut = 3 + 4 - 1 = 6
Slide 31
Slide 31 text
Observation 1
The network flow sent across any cut is equal to the
amount reaching sink.
1
3
2 4
5
6
3/3
3/8
4/4
2/2
1/4
4/4
2/3
total flow = 6, flow on cut = 4 + 2= 6
Slide 32
Slide 32 text
Observation 1
The network flow sent across any cut is equal to the
amount reaching sink.
1
3
2 4
5
6
3/3
3/8
4/4
2/2
1/4
4/4
2/3
total flow = 6, flow on cut = 4 + 2= 6
Slide 33
Slide 33 text
Observation 1
The network flow sent across any cut is equal to the
amount reaching sink.
1
3
2 4
5
6
3/3
3/8
4/4
2/2
1/4
4/4
2/3
total flow = 6, flow on cut = 4 + 2= 6
Slide 34
Slide 34 text
Observation 1
The network flow sent across any cut is equal to the
amount reaching sink.
1
3
2 4
5
6
3/3
3/8
4/4
2/2
1/4
4/4
2/3
total flow = 6, flow on cut = 4 + 2= 6
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Observation 2
Then the value of the flow is at most the capacity of
any cut.
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It’s trivial!
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Observation 2
Then the value of the flow is at most the capacity of
any cut.
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8
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3
It’s trivial!
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Observation 3
Let f be a flow, and let (S,T) be an s-t cut whose
capacity equals the value of f.
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2 4
5
6
3/3
3/8
4/4
2/2
1/4
4/4
2/3
f is the maximum flow
(S,T) is the minimum cut
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Observation 3
Let f be a flow, and let (S,T) be an s-t cut whose
capacity equals the value of f.
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2 4
5
6
3/3
3/8
4/4
2/2
1/4
4/4
2/3
f is the maximum flow
(S,T) is the minimum cut