1. Introduction to Maximum Flows
Lecture 13
Introduction to Maximum Flows

2. Flow Network

8. The Ford Fulkerson Maximum Flow Algorithm
Begin x := 0;
create the residual network G(x);
while there is some directed path from s to t in G(x) do
begin
let P be a path from s to t in G(x);
∆:= δ(P);
send ∆ units of flow along P;
update the r's;
End
end {the flow x is now maximum}.

9. The Ford-Fulkerson Augmenting Path Algorithm for the Maximum Flow Problem
Obtain a network, and use the same network to illustrate the shortest path problem for communication networks, the max flow problem, the minimum cost flow problem, and the multicommodity flow problem. This will be a very efficient way of introducing the four problems. (Perhaps under 10 minutes of class time.)

10. Ford-Fulkerson Max Flow4 2 5 1 3 1 1 2 2 s 4 t 3 2 1 3
This is the original network, and the original residual network.

11. Ford-Fulkerson Max Flow4 2 5 1 3 1 1 2 2 s 4 t 3 2 1 3
Find any s-t path in G(x)

12. Ford-Fulkerson Max Flow4 2 5 1 3 1 1 2 1 2 s 4 t 1 2 3 2 1 1 1 3
Determine the capacity D of the path.
Send D units of flow in the path. Update residual capacities.

13. Ford-Fulkerson Max Flow4 2 5 1 3 1 1 2 1 2 s 4 t 1 2 3 2 1 1 1 3
Find any s-t path

14. Ford-Fulkerson Max Flow4 2 5 1 3 1 1 2 1 1 1 s 4 t 1 1 2 3 2 2 1 1 1 1 1 3
Determine the capacity D of the path.
Send D units of flow in the path. Update residual capacities.

15. Ford-Fulkerson Max Flow4 2 5 1 3 1 1 2 1 1 1 s 4 t 1 1 2 3 2 2 1 1 1 1 1 3
Find any s-t path

16. Ford-Fulkerson Max Flow4 2 5 1 3 1 1 1 1 1 s 4 t 1 2 2 1 3 2 1 1 1 1 1 3
Determine the capacity D of the path.
Send D units of flow in the path. Update residual capacities.

17. Ford-Fulkerson Max Flow4 2 5 1 3 1 1 2 1 1 1 s 4 t 2 1 1 2 2 3 1 1 1 1 1 3
Find any s-t path

18. Ford-Fulkerson Max Flow4 2 5 1 3 1 1 2 1 1 1 s 4 t 2 1 1 2 2 1 1 1 1 2 1 1 2 3
Determine the capacity D of the path.
Send D units of flow in the path. Update residual capacities.

19. Ford-Fulkerson Max Flow4 2 5 1 3 1 1 2 1 1 1 s 4 t 1 2 1 2 1 2 1 1 2 1 2 1 3
Find any s-t path

20. Ford-Fulkerson Max Flow4 3 2 5 1 1 2 3 1 1 1 1 s 4 t 2 1 1 2 1 2 1 1 2 1 1 2 3
Determine the capacity D of the path.
Send D units of flow in the path. Update residual capacities.

21. Ford-Fulkerson Max Flow3 4 2 5 1 1 2 3 1 1 1 1 s 4 t 1 2 1 2 1 2 1 1 2 1 2 1 3
There is no s-t path in the residual network. This flow is optimal

22. Ford-Fulkerson Max Flow4 3 2 2 5 5 1 1 2 3 1 1 1 1 s s 4 4 t 2 1 2 1 1 2 1 1 1 2 1 2 3 3
These are the nodes that are reachable from node s.

23. Ford-Fulkerson Max Flow1 2 s 4 5 3 t
Here is the optimal flow

28. Review for cutset theorem

37. Quiz Sample

44. Polynomial-time