Complexity of FordFulkerson

1
Complexity of Ford-Fulkerson

• Let U max (i,j) in A uij.
• If S s and T N\s, then uS,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
3
An Augmenting Path
(0,106)
(1,106)
2
1
5
(1,1)
s
t
3
(1,106)
(0,106)
v 1
4
Residual Network
106
106-1
2
1
5
1
s
t
0
1
1
106
3
106-1
5
An Augmenting Path in the Residual Network
106
106-1
2
1
5
1
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
5
1
1
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
5
1
1
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)
• Goldbergs algorithm (preflow-push) with
  highest-label implementation.
• O(n2m1/2)