1. Maximum Flow Solutions
ENGM 435/535
Maximum Flow Solutions

2. Maximum Flow Models 2 4 6 1 3 5 (4,4) (6,8) (4,10) (2,2) (0,3) (0,8)
(Flow, Capacity)
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(0,3)
(0,8) 6 1
(3,6)
(5,7)
(3,3) 3 5

3. Maximum Flow Models Maximal Flow 2 4 6 1 3 5 S S
(Flow, Capacity)
[External Flow]
Maximal Flow
Capacity is only relevant parameter.
Find maximal flow from source to sink.
(4,4) 2 4
(6,8)
(4,10)
[-M]
[M]
(2,2)
(0,3)
(0,8) S 6 S 1
(3,6)
(5,7)
(3,3) 3 5

4. Maximum Flow
Find a flow augmenting path defined by a sequence of arcs P =(k1, k2,. . . kp)
Determine the maximum flow increase along the path
Change the flow in the arcs on the path
Repeat until no flow augmenting paths can be found

5. Maximum Flow Find an augmenting path
Determine the maximum flow augmentation possible
Augment flow by that amount

6. Maximum Flow Models 2 4 6 1 3 5 (0,4) (0,8) (0,10) (0,2) (0,3) (0,8)
Find a path top to bottom that has
Additional capacity. Increase flow to
Available capacity
(Flow, Capacity)
(0,4) 2 4
(0,8)
(0,10)
(0,2)
(0,3)
(0,8) 6 1
(0,6)
(0,7)
(0,3) 3 5

7. Augmented Path 2 4 6 1 3 5 (4,4) (4,8) (4,10) (0,2) (0,3) (0,8) (0,6)
(Flow, Capacity)
(4,4) 2 4
(4,8)
(4,10)
(0,2)
(4)
(0,3)
(0,8) 6 1
(0,6)
(0,7)
(0,3) 3 5

8. Augmented Path 2 4 6 1 3 5 (4,4) (4,8) (4,10) (0,2) (0,3) (0,8) (0,6)
(Flow, Capacity)
(4,4) 2 4
(4,8)
(4,10)
(0,2)
(4)
(0,3)
(0,8) 6 1
(0,6)
(0,7)
(0,3) 3 5

9. Augmented Path 2 4 6 1 3 5 (4,4) (4,8) (4,10) (0,2) (0,3) (0,8) (0,6)
(Flow, Capacity)
(4,4) 2 4
(4,8)
(4,10)
(0,2)
(4)
(0,3)
(0,8) 6 1
(0,6)
(0,7)
(0,3) 3 5

10. Augmented Path 2 4 6 1 3 5 (4,4) (6,8) (4,10) (2,2) (0,3) (0,8) (0,6)
(Flow, Capacity)
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(6)
(0,3)
(0,8) 6 1
(0,6)
(2,7)
(0,3) 3 5

11. Augmented Path 2 4 6 1 3 5 (4,4) (6,8) (4,10) (2,2) (0,3) (0,8) (0,6)
(Flow, Capacity)
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(6)
(0,3)
(0,8) 6 1
(0,6)
(2,7)
(0,3) 3 5

12. Augmented Path 2 4 6 1 3 5 (4,4) (6,8) (4,10) (2,2) (0,3) (0,8) (0,6)
(Flow, Capacity)
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(6)
(0,3)
(0,8) 6 1
(0,6)
(2,7)
(0,3) 3 5

13. Augmented Path 2 4 6 1 3 5 (4,4) (6,8) (4,10) (2,2) (0,3) (0,8) (0,6)
(Flow, Capacity)
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(6)
(0,3)
(0,8) 6 1
(0,6)
(2,7)
(0,3) 3 5

14. Augmented Path 2 4 6 1 3 5 (4,4) (6,8) (4,10) (2,2) (0,3) (0,8) (3,6)
(Flow, Capacity)
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(9)
(0,3)
(0,8) 6 1
(3,6)
(5,7)
(3,3) 3 5

15. Augmented Path 2 4 6 1 3 5 (4,4) (6,8) (4,10) (2,2) (0,3) (0,8) (3,6)
(Flow, Capacity)
Arc 2-4 at capacity
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(9)
(0,3)
(0,8) 6 1
(3,6)
(5,7)
(3,3) 3 5

16. Augmented Path 2 4 6 1 3 5 (4,4) (6,8) (4,10) (2,2) (0,3) (0,8) (3,6)
(Flow, Capacity)
Arc 2-4 at capacity
Arc 2-5 at capacity
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(9)
(0,3)
(0,8) 6 1
(3,6)
(5,7)
(3,3) 3 5

17. Augmented Path 2 4 6 1 3 5 (4,4) (6,8) (4,10) (2,2) (0,3) (0,8) (3,6)
(Flow, Capacity)
Arc 2-4 at capacity
Arc 2-5 at capacity
Arc 3-5 at capacity
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(9)
(0,3)
(0,8) 6 1
(3,6)
(5,7)
(3,3) 3 5

18. Augmented Path 2 4 6 1 3 5 (4,4) (6,8) (4,10) (2,2) (0,3) (0,8) (3,6)
(Flow, Capacity)
No other path exists start to end
that has additional capacity
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(9)
(0,3)
(0,8) 6 1
(3,6)
(5,7)
(3,3) 3 5

19. Augmented Path 2 4 6 1 3 5 (4,4) (6,8) (4,10) (2,2) (0,3) (0,8) (3,6)
(Flow, Capacity)
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(9)
(0,3)
(0,8) 6 1
(3,6)
(5,7)
(3,3) 3 5

20. Minimum Cut Algorithm Find all possible cuts source to sink
Find the cut that has minimal capacity
Minimal capacity cut = maximum flow

21. Minimum Cut Algorithm 2 4 6 1 3 5 (4,4) (6,8) (4,10) (2,2) (0,3) (0,8)
(Flow, Capacity)
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(0,3)
(0,8) 6 1
(3,6)
(5,7)
(3,3) 3 5
(Capacity = 14)

22. Minimum Cut Algorithm 2 4 6 1 3 5 (4,4) (6,8) (4,10) (2,2) (0,3) (0,8)
(Flow, Capacity)
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(0,3)
(0,8) 6 1
(3,6)
(5,7)
(3,3) 3 5
(Capacity = 14)

23. Minimum Cut Algorithm 2 4 6 1 3 5 (4,4) (6,8) (4,10) (2,2) (0,3) (0,8)
(Flow, Capacity)
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(0,3)
(0,8) 6 1
(3,6)
(5,7)
(3,3) 3 5
(Capacity = 11)

24. Minimum Cut Algorithm 2 4 6 1 3 5 (4,4) (6,8) (4,10) (2,2) (0,3) (0,8)
(Flow, Capacity)
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(0,3)
(0,8) 6 1
(3,6)
(5,7)
(3,3) 3 5
(Capacity = 17)

25. Maximum Flow Models 2 4 6 1 3 5 (4,4) (6,8) (4,10) (2,2) (0,3) (0,8)
(Flow, Capacity)
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(0,3)
(0,8) 6 1
(3,6)
(5,7)
(3,3) 3 5
(Capacity = 9)

26. Maximum Flow Models 2 4 6 1 3 5 (4,4) (6,8) (4,10) (2,2) (0,3) (0,8)
(Flow, Capacity)
(4,4) 2 4
(6,8)
(4,10)
(2,2)
(0,3)
(0,8) 6 1
(3,6)
(5,7)
(3,3) 3 5
(Capacity = 9)