max flow min cut. theorem the maximum value of an st-flow in a digraph equals the minimum capacity...
TRANSCRIPT
![Page 1: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/1.jpg)
Max Flow Min Cut
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Theorem
The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut.
Theorem
If every arc has integer capacity, then in a maximum flow every arc has integer flow.
![Page 3: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/3.jpg)
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Find a maximum st-flow and a minimum st-cut
s t
![Page 5: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/5.jpg)
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Algorithm
s t1
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7
![Page 6: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/6.jpg)
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Algorithm
s t
}{sS
1
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2
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9
7
0f
![Page 7: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/7.jpg)
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Algorithm
s t
}{sS }4,{sS
1
3
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2
4
8
6
9
7
0f
![Page 8: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/8.jpg)
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Algorithm
s t
}{sS }4,{sS
1
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}7,4,{sS 0f
![Page 9: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/9.jpg)
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Algorithm
s t
}{sS }4,{sS
1
3
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9
7
}7,4,{sS 0f
}9,7,4,{sS
![Page 10: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/10.jpg)
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Algorithm
s t
}{sS }4,{sS
1
3
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2
4
8
6
9
7
}7,4,{sS 0f
}9,7,4,{sS },9,7,4,{ tsS
![Page 11: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/11.jpg)
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Algorithm
s t
}{sS }4,{sS
1
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9
7
}7,4,{sS 0f
}9,7,4,{sS },9,7,4,{ tsS
STOP St
![Page 12: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/12.jpg)
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Algorithm
s t
}{sS }4,{sS
1
3
5
2
4
8
6
9
7
}7,4,{sS 0f
}9,7,4,{sS },9,7,4,{ tsS
STOP St
![Page 13: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/13.jpg)
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Algorithm
s t
}{sS }4,{sS
1
3
5
2
4
8
6
9
7
}7,4,{sS
}9,7,4,{sS },9,7,4,{ tsS
STOP St 3),9()9,7(
)7,4()4,(
tff
fsf
![Page 14: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/14.jpg)
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Algorithm
s t
}{sS }4,{sS
1
3
5
2
4
8
6
9
7
}7,4,{sS
}9,7,4,{sS },9,7,4,{ tsS
STOP St 3),9()9,7(
)7,4()4,(
tff
fsf
![Page 15: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/15.jpg)
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Algorithm
s t1
3
5
2
4
8
6
9
7
3),9()9,7(
)7,4()4,(
tff
fsf
![Page 16: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/16.jpg)
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Algorithm
s t1
3
5
2
4
8
6
9
7
3),9()9,7(
)7,4()4,(
tff
fsf}{sS }1,{sS }2,1,{sS
},6,8,5,2,1,{ tsS
STOP St
![Page 17: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/17.jpg)
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Algorithm
s t1
3
5
2
4
8
6
9
7
3),9()9,7(
)7,4()4,(
tff
fsf}{sS }1,{sS }2,1,{sS
},6,8,5,2,1,{ tsS
STOP St
1),6(...
)2,1()1,(
tf
fsf
![Page 18: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/18.jpg)
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Algorithm
s t1
3
5
2
4
8
6
9
7
3),9()9,7(
)7,4()4,(
tff
fsf}{sS }1,{sS }2,1,{sS
},6,8,5,2,1,{ tsS
STOP St
1),6(...
)2,1()1,(
tf
fsf
![Page 19: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/19.jpg)
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Algorithm
s t1
3
5
2
4
8
6
9
7
3),9()9,7(
)7,4()4,(
tff
fsf
1),6(...
)2,1()1,(
tf
fsf
![Page 20: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/20.jpg)
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Algorithm
s t1
3
5
2
4
8
6
9
7
3),9()9,7(
)7,4()4,(
tff
fsf
1),6(...
)2,1()1,(
tf
fsf
}{sS
![Page 21: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/21.jpg)
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Algorithm
s t1
3
5
2
4
8
6
9
7
3),9()9,7(
)7,4()4,(
tff
fsf
1),6(...
)2,1()1,(
tf
fsf
}{sS }6,{sS
![Page 22: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/22.jpg)
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Algorithm
s t1
3
5
2
4
8
6
9
7
3),9()9,7(
)7,4()4,(
tff
fsf
1),6(...
)2,1()1,(
tf
fsf
}{sS }6,{sS
},6,{ tsS STOP St
![Page 23: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/23.jpg)
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Algorithm
s t1
3
5
2
4
8
6
9
7
3),9()9,7(
)7,4()4,(
tff
fsf
1),6(...
)2,1()1,(
tf
fsf
}{sS }6,{sS
},6,{ tsS STOP St
2),6(,1)6,( tfsf
![Page 24: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/24.jpg)
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Algorithm
s t1
3
5
2
4
8
6
9
7
3),9()9,7(
)7,4()4,(
tff
fsf
1),6(...
)2,1()1,(
tf
fsf
}{sS }6,{sS
},6,{ tsS STOP St
2),6(,1)6,( tfsf
![Page 25: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/25.jpg)
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21
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Algorithm
s t1
3
5
2
4
8
6
9
7
3),9()9,7(
)7,4()4,(
tff
fsf
1),6(...
)2,1()1,(
tf
fsf
}{sS }6,{sS
},6,{ tsS STOP St
2),6(,1)6,( tfsf
5)( fv
![Page 26: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/26.jpg)
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Minimum Cut
s t1
3
5
2
4
8
6
9
7
5)( fv
![Page 27: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/27.jpg)
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Minimum Cut
s t1
3
5
2
4
8
6
9
7
5)( fv
![Page 28: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/28.jpg)
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Minimum Cut
s t1
3
5
2
4
8
6
9
7
5)( fv5113),( SSc
![Page 29: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/29.jpg)
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1
31
1
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1
1
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3
s t1
3
5
2
4
8
6
9
7
So f is a max flow
![Page 30: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/30.jpg)
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Algorithm
s t
}{sS }4,{sS
1
3
5
2
4
8
6
9
7
}7,4,{sS 0f
![Page 31: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/31.jpg)
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Algorithm
s t
}{sS }4,{sS
1
3
5
2
4
8
6
9
7
}7,4,{sS 0f
},6,8,7,4,{ tsS
![Page 32: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/32.jpg)
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Algorithm
s t
}{sS }4,{sS
1
3
5
2
4
8
6
9
7
}7,4,{sS
},6,8,7,4,{ tsS
3),6()6,8()8,9(
)9,7()7,4()4,(
tfff
ffsf
![Page 33: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/33.jpg)
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Algorithm
s t1
3
5
2
4
8
6
9
7
3),6()6,8()8,9(
)9,7()7,4()4,(
tfff
ffsf
![Page 34: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/34.jpg)
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1
2 3
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Algorithm
s t1
3
5
2
4
8
6
9
7
3),6()6,8()8,9(
)9,7()7,4()4,(
tfff
ffsf}{sS }6,{sS
}8,6,{sS
},9,8,6,{ tsS
![Page 35: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/35.jpg)
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1
2 3
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Algorithm
s t1
3
5
2
4
8
6
9
7
3),6()6,8()8,9(
)9,7()7,4()4,(
tfff
ffsf}{sS }6,{sS
}8,6,{sS
},9,8,6,{ tsS
2)9,8()8,6(
1),9()6,(
ff
tfsf
![Page 36: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/36.jpg)
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2
3
2
1
2 32
323
31
1
11
1 2
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Algorithm
s t1
3
5
2
4
8
6
9
7
3),6()6,8()8,9(
)9,7()7,4()4,(
tfff
ffsf}{sS }6,{sS
}8,6,{sS
},9,8,6,{ tsS
2)9,8()8,6(
1),9()6,(
ff
tfsf
![Page 37: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/37.jpg)
Hall’s Theorem from
Max Flow Min Cut
![Page 38: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/38.jpg)
![Page 39: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/39.jpg)
1 11
1
1
Direct all edges from s to t and assign all arcs unit capacity
Adds and t, adjacent to all of A and B respectively.
![Page 40: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/40.jpg)
We have to show that Hall’s Condition XX )(gives a 1-factor.
![Page 41: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/41.jpg)
A flow of value is enough to guarantee a 1-factor.
A
![Page 42: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/42.jpg)
So all we have to do is show for each cut S.
ASSc ),(
![Page 43: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/43.jpg)
So all we have to do is show for each cut S.
ASSc ),(
![Page 44: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/44.jpg)
![Page 45: Max Flow Min Cut. Theorem The maximum value of an st-flow in a digraph equals the minimum capacity of an st-cut. Theorem If every arc has integer capacity,](https://reader037.vdocuments.net/reader037/viewer/2022103112/551a9336550346b52d8b5f00/html5/thumbnails/45.jpg)
),()\,()\,(),( tSBeSBSAeSAseSSc )\,(\ SBSAeSBSA
)(\ SASA
ASASA \