Complexity of Ford-Fulkerson

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

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

4. Residual Network 106 106-1 2 1 1 5 s t 0 1 1 106 3 106-1

5. An Augmenting Path in the Residual Network 106 106-1 2 1 1 5 s t 0 1 1 106 3 106-1

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

7. Updated Residual Network 106 -1 106-1 2 1 1 1 5 s t 1 0 1 1 106 -1 3 106-1

8. Next Augmenting Path in the Residual Network 106 -1 106-1 2 1 1 1 5 s t 1 0 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)