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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,

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Presentation on theme: "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,"— Presentation transcript:

1 Max Flow Min Cut

2 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.

3

4 Find a maximum st-flow and a minimum st-cut s t

5 Algorithm s t

6 s t

7 s t

8 s t

9 s t

10 s t

11 s t STOP

12 Algorithm s t STOP

13 Algorithm s t STOP

14 Algorithm s t STOP

15 Algorithm s t

16 s t STOP

17 Algorithm s t STOP

18 Algorithm s t STOP

19 Algorithm s t

20 s t

21 s t

22 s t STOP

23 Algorithm s t STOP

24 Algorithm s t STOP

25 Algorithm s t STOP

26 Minimum Cut s t

27 s t

28 s t

29 s t So f is a max flow

30 Algorithm s t

31 s t

32 s t

33 s t

34 s t

35 s t

36 s t

37 Hall’s Theorem from Max Flow Min Cut

38

39 11111 Direct all edges from s to t and assign all arcs unit capacity Adds and t, adjacent to all of A and B respectively.

40 We have to show that Hall’s Condition gives a 1-factor.

41 A flow of value is enough to guarantee a 1-factor.

42 So all we have to do is show for each cut S.

43

44

45


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