1. ENGM 631 Maximum Flow Solutions

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

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

4. Maximum Flow 1.Find a flow augmenting path defined by a sequence of arcs P =(k1, k2,.v.v.vkp) 2.Determine the maximum flow increase along the path 3.Change the flow in the arcs on the path 4.Repeat until no flow augmenting paths can be found

5. Maximum Flow 1.Find an augmenting path 2.Determine the maximum flow augmentation possible 3.Augment flow by that amount

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

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

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

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

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

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

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

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

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

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

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

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

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

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

20. Minimum Cut Algorithm 1.Find all possible cuts source to sink 2.Find the cut that has minimal capacity 3.Minimal capacity cut = maximum flow

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

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

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

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

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

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

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