Maximum Flow Solutions

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Presentation transcript:

Maximum Flow Solutions ENGM 435/535 Maximum Flow Solutions

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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)

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)

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)

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)

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)