ENGM 792 Network Flow Programming

1. ENGM 792Network Flow Programming Maximum Flow Solutions

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

3. Maximum Flow Models (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) 6 S S 1 (5,7) (3,6) (3,3) 5 3

4. Maximum Flow • Find a flow augmenting path defined by a sequence of arcs P =(k1, k2,.v.v.vkp) • 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 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 (0,7) (0,6) (0,3) 5 3

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

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

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

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

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

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

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

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

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

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

17. Augmented Path (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) (9) (0,3) (0,8) 6 1 (3,7) (3,6) (3,3) 5 3

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

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

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 (Flow, Capacity) (4,4) 2 4 (6,8) (4,10) (2,2) (0,3) (0,8) 6 1 (5,7) (3,6) (3,3) 5 3 (Capacity = 14)

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

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

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

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

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

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