1. Reading before Lecture 13
Network Flow

3. Flow Decomposition and Transformations

4. j

13. The Flow Decomposition Algorithm
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.)

14. begin
Initialize
while y   do
Select(s, y)
Search(s, y)
if a cycle C is found then do
let D = Capacity(C, y)
Add Flow(D, C) to cycle flows
Subtract Flow(D, C) from y.
end
if a path P is found then do
let D = Capacity(P, y)
Add Flow(D, P) to path flows
Subtract Flow(D, P) from y.

15. The initial flow 7 2 4 4 3 3 8 2 1 6 -2 4 4 2 3 5 5 9
A supply node
The flow x
A demand node
A balanced node

16. Find a Path or Cycle 7 2 4 4 3 3 8 2 1 1 6 -2 4 4
Carry out a depth first search until a cycle C or a path P is found
Select a supply node 2 3 5 5 9
Determine the capacity
The capacity is 2.

17. Update the decomposition5 7 2 4 2 4 3 3 8 6 2 1 1 6 -2 4 4
Add the cycle flow to the decomposition 2 3 5 5 9 7
update the current flow
2 units around
cycle flows
path flows

18. Find the next path or cycle5 2 4 3 2 3 6
Start at a supply node and find the next cycle or path 2 1 1 6 -2 4 4 3 5 5 7
2 units around
cycle flows
path flows

19. update the decomposition and current flow2 5 2 4 3 2 3 6 3
update the current flow and decomposition 2 1 1 6 -2 4 4 3 5 5 7 4
2 units around
3 units around
cycle flows
path flows

20. Find the next flow or cycle2 2 4 3 2
start with a supply node and find the next flow or cycle. 3 2 1 1 6 -2 4 4 3 5 5 4
The capacity of is 3
2 units around
3 units around
cycle flows
path flows

21. update the current flow and decomposition 2 1 1 6 -2 4 1 43 2 3
update the current flow and decomposition 2 1 1 6 -2 4 1 4 3 5 5 1 4
2 units around
3 units around
3 units around
cycle flows
path flows

22. Find the next flow or cycle
start with a supply node and find the next flow or cycle. 2 2 4 3 2 2 1 1 6 -2 1 4
The capacity of is 2 3 5 5 1 4
2 units around
3 units around
3 units around
cycle flows
path flows

23. update the current flow and decomposition 2 2 4 3 1 22 2 4 3 1 2 2 1 1 6 -2 1 4 3 5 5 1 4
2 units around
3 units around
2 units in
3 units around
cycle flows
path flows

24. Find the next path or cycle
start with any node incident to an arc with flow and find the next flow or cycle. 2 4 1 1 6 -2 1 4 3 3 5 5 4 1
2 units around
3 units around
2 units in
3 units around
cycle flows
path flows

25. Update 2 4 1 1 6 -2 1 4 3 3 5 5 4 4 1 cycle flows path flows2 4 1 1 6 -2 1 4 3 3 5 5 4 4 1
2 units around
3 units around
2 units in
3 units around
1 units around
cycle flows
path flows

26. The Final Decomposition2 4 1 6 -2 4 3 5 4
2 units around
3 units around
3 units around
2 units in
1 units around
4 units around 5-6-5
cycle flows
path flows

36. (yki + lki)

37. bi 1 1
bj+uij
-uij -1 -1
-uij -1 -1
-uij
xij
xij
sij bi
bj+uij bj bi

38. Exercise Transform the following network for removing arc capacities.
(5,10)
(6,30)
(cij, uij)
(4,20)
(1,10)

41. Homework Readings Section 2.4 Section 3.5 Exercises Exercise 2.42