1. Complexity of Ford-Fulkerson
Let U = max {(i,j) in A} uij.
If S = {s} and T = N\{s}, then u[S,T] is at most nU.
The maximum flow is at most nU.
At most nU augmentations.
Each iteration of the inner while loop is O(m):
Each arc is inspected at most once
Finding  is O(n)
Updating the flow on P is O(n)
Complexity is O(nmU).

2. Pathological Example s 1 2 3 5
(0,106)
(0,1) t

3. An Augmenting Path 2 1 5 3 (1,106) (0,106) (1,1) s t (0,106) (1,106)
v = 1

4. Residual Network
106-1 2
106 1 1 5 s 1 t 1
106 3
106-1

5. An Augmenting Path in the Residual Network
106-1 2
106 1 1 5 s 1 t 1
106 3
106-1

6. Updated Flow
(1,106) 2
(1,106) 1 5
(0,1) s t 3
(1,106)
(1,106)
v = 2

7. Updated Residual Network
106-1 2
106 -1 1 1 1 5 s 1 t 1 1
106 -1 3
106-1

8. Next Augmenting Path in the Residual Network
106-1 2
106 -1 1 1 1 5 s 1 t 1 1
106 -1 3
106-1
This will take 2 million iterations to find the maximum flow!

9. Polynomial Max Flow Algorithms (Chapter 7)
Always augment along the shortest augmenting path in the residual network.
O(n2m)
Always augment along the maximum-capacity augmenting path in the residual network.
O(nm log U)
Goldberg’s algorithm (preflow-push) with highest-label implementation.
O(n2m1/2)