1. Max-flow, Min-cut
Network flow

2. Max-flow
Maximize the total amount of flow from s to t subject to two constraints
Flow on edge e doesn’t exceed c(e)
For every node v ≠ {s, t}, incoming flow is equal to outgoing flow

3. Max-flow: Ford-Fulkerson
Find paths from s to t using depth first search
Find paths using the residual graph G’

4. Ford-Fulkerson: example4 10
0|4 2 2
0|10
0|2
0|2 7 10 7 s t
0|7
0|10
0|7 s t 2 2
0|2
0|2 6 10
0|10
0|6 G G
Depth first search
f = min(4,10)=4
0|4
0|10
4|4
4|10
0|2
0|2
0|2
0|2
0|7
0|10
0|7
0|7
0|10
0|7 s t s t
0|2
0|2
0|2
0|2
0|6
0|6
0|10
0|10 G G

5. Ford-Fulkerson: DFS s t s t s t s t G G G G f = min(7,10,7)=7
4|4
4|10
4|4
4|10
0|2
0|2
0|2
0|2
0|7
0|10
0|7
7|7
7|10
7|7 s t s t
0|2
0|2
0|2
0|2
0|6
0|10
0|6
0|10 G G
f = min(10,6)=6
4|4
4|10
4|4
4|10
0|2
0|2
0|2
0|2
7|7
7|10
7|7
7|7
7|10
7|7 s t s t
0|2
0|2
0|2
0|2
0|10
0|6
6|6
6|10 G G

6. Ford-Fulkerson: Residual graph
4|4
4|10 ? ?
0|2
0|2 ? ?
7|7
7|10
7|7 ? ? ? s t s t
0|2
0|2 ? ?
6|6 ?
6|10 ? G G’ G G’
4|4
4|10 4 ?
0|2
0|2 ?
7|7
7|10
7|7 s t ? ? ? s t
0|2
0|2 ? ?
6|6
6|10 ? ?

7. Ford-Fulkerson: Residual graph
4|4
4|10 4 4
0|2
0|2 ? 6
7|7
7|10
7|7 s t ? ? ? s t
0|2
0|2 ? ?
6|6
6|10 ? ? G G’
4|4
4|10 4 4
0|2
0|2 ? 6
7|7
7|10
7|7 s t ? ? s t 7
0|2
0|2 ? ?
6|6
6|10 ? ? G G’

8. Ford-Fulkerson: Residual graph
4|4
4|10 4 4
0|2
0|2 2 6
7|7
7|10
7|7 s t ? ? s t 7
0|2
0|2 ? ?
6|6
6|10 ? ? G G’
4|4
4|10 4 4
0|2
0|2 2 6
7|7
7|10
7|7 s t ? ? s t 7
0|2
0|2 ? 4 ?
6|10
6|6 6 ? G G’

9. Ford-Fulkerson: Residual graph
Keep work on for the rest of edges
4|4
4|10 4 4
0|2
0|2 2 2 6
7|7
7|10
7|7 s t 3 s t 7 7 7
0|2
0|2 2 4 2
6|6
6|10 6 6 G G’ 4 4 2 2 6 3 s t
Remove “0” edge (optional) 7 7 4 2 6 2 6 G’

10. Ford-Fulkerson: Residual graph
Any more paths? 4 4 2 2 6 3 s t 7 7 2 4 2 6 6 G’
f = min(6,2,3,2,4)=2 4 4
4|4 2
6|10 2 6
2|2
0|2 3 s t
7|7
5|10
7|7 s t 7 7 2 4 2 6
0|2
2|2 6
6|6
8|10 G’ G

11. Max flow = 6 + 7 +6 = 19 s t G 4|4 6|10 2|2 0|2 7|7 5|10 7|7 0|2 2|2
8|10
6|6 G

12. Edmonds-Karp
Edmonds-Karp = Ford-Fulkerson + “Choose the augmenting path with the smallest number of edges” or “Choose the augmenting path with the largest bottle neck value”

13. Edmonds-Karp vs Ford-Fulkerson
0|4
0|10
0|2
0|2
0|7
0|10
0|7 s t
0|2
0|2
0|6
Edmonds-Karp
Ford-Fulkerson G
Ford-Fulkerson
0|4
0|10
0|2
0|2
0|4
0|10
0|2
0|2
0|7
0|10
0|7 s t
0|7
0|10
0|7 s t
0|2
0|2
0|10
0|6
0|2
0|2
0|10
0|6 G G
Which one is the valid first choice of Edmonds-Karp?
Which one is the valid first choice of Ford-Fulkerson?

14. Min cut
We want to remove some edges from the graph such that after removing the edges, there is no path from s to t
The cost of removing e is equal to its capacity c(e)
The minimum cut problem is to find a cut with minimum total cost

15. Min cut: approach “Subtract” the max-flow from the original graph
Mark all nodes reachable from s. Call the set of reachable nodes A
Now separate these nodes from the others
Cut edges going from A to V − A

16. Min cut: example s t s t s t G G max flow4 10 4 6 2 2 2 7 10 7 7 s t 5 7 s t 2 2 2 6 8 10 6 G
G max flow s t
G – G max flow = residual graph

17. Min cut: example s t s t s t G G max flow4 10 4 6 2 2 2 7 10 7 7 s t 5 7 s t 2 2 2 6 8 10 6 G
G max flow s t
G – G max flow = residual graph

18. Min cut: example s t s t s t G G max flow4 10 4 6 2 2 2 7 10 7 7 s t 5 7 s t 2 2 2 6 8 10 6 G
G max flow s t
G – G max flow = residual graph

19. Min cut: example s t s t s t G G max flow4 10 4 6 2 2 2 7 10 7 7 s t 5 7 s t 2 2 2 6 8 10 6 G
G max flow s t
G – G max flow = residual graph

20. Min cut: example s t s t s t G G max flow4 10 4 6 2 2 2 7 10 7 7 s t 5 7 s t 2 2 2 6 8 10 6 G
G max flow s t
G – G max flow = residual graph

21. Min cut: example s t s t s t G G max flow4 10 4 6 2 2 2 7 10 7 7 s t 5 7 s t 2 2 2 6 8 10 6 G
G max flow s t
G – G max flow = residual graph

22. Min cut: example s t s t s t G G max flow4 10 4 6 2 2 2 7 10 7 7 s t 5 7 s t 2 2 2 6 8 10 6 G
G max flow s t
G – G max flow = residual graph

23. Min cut: example s t s t s t G G max flow4 10 4 6 2 2 2 7 10 7 7 s t 5 7 s t 2 2 2 6 8 10 6 G
G max flow s t
G – G max flow = residual graph

24. Nodes reachable from s (A)t

25. Cut edges come from V - A s t s t G G – G max flow = residual graph4 10 2 2 7 10 7 s t s t 2 2 6 10 G
G – G max flow = residual graph
Cost of min cut = = 19 = max flow value