1. Network Optimization
Flow Decomposition

2. The initial flow 7 2 4 4 3 3 8 2 -2 1 6 4 4 2 3 5 5 9
A deficit node (more flow leaving than entering).
An excess node (more flow entering than leaving)
A balanced node (flow in = flow out)

3. Find a Path or Cycle W Select a node with deficit if there is one. 7 24 4 3 3 8 2 1 1 6 -2 4 4 2
Carry out a depth first search. Stop when a node with excess is reached or when there is a cycle. 3 5 5 9 2 4 5 3 1
Determine the capacity of the walk W.
The capacity of is 2.

4. Updates Add the flow in W to the decomposition. 7 5 2 4 4 2 3 3 6 8 21 1 6 -2 4 4
Subtract the flow in W from the current flow. 2 3 5 5 9 7
2 units around
cycle flows
path flows

5. Find a path or cycle W Select a node with deficit if there is one. 5 24 2 3 3 6 2 1 1 6 -2 4 4
Carry out a dfs.
Determine the capacity of W. 3 5 5 7 2 4 5 3
The capacity of is 3.

6. Updates Add the cycle flow to the decomposition 2 5 2 4 2 3 3 6 3 2 16 3 2 1 1 6 -2 4 4
update the current flow 3 5 5 4 7
2 units around
3 units around
cycle flows
path flows

7. Find a path or cycle using dfs
Select a node with deficit if there is one. 2 2 4 2 3 3 2 1 1 6 -2 4 4
Carry out a dfs.
Determine the capacity of W. 3 5 5 4 6 2 3 -2 1 4
capacity of a path = min {arc capacity, excess, deficit} = 2

8. Updates Add the path flow to the decomposition 2 2 4 2 1 3 3 2 1 1 6 62 2 4 2 1 3 3 2 1 1 6 6 -2 4 4
update the current flow 3 5 5 4
2 units around
2 units in
3 units around
cycle flows
path flows

9. Find a path or cycle using dfs
Select a node with deficit if there is one. Otherwise, select any node with flow leaving. 2 4 1 3 1 1 6 6 4 4 3 3 5 5 4
Carry out a dfs. 5 4 6 3
Determine the capacity of W.
The capacity is 1

10. Updates Add the cycle flow to the decomposition 2 4 1 3 1 1 6 6 3 4 43 1 1 6 6 3 4 4
update the current flow 3 5 4 5 4 3
2 units around
2 units in
3 units around
1 unit around
cycle flows
path flows

11. Find a path or cycle using dfs
Select a node with deficit if there is one. Otherwise, select any node with flow leaving. 2 4 3 1 1 6 6 3 4 3 3 5 4 3
Carry out a dfs.
Determine the capacity of W.
The capacity of is 3

12. Updates Add the cycle flow to the decomposition 2 4 3 1 1 6 6 3 4
update the current flow 3 5 4 3
2 units around
3 units around
2 units in
1 unit around
3 units around
cycle flows
path flows

13. Find a path or cycle using dfs
Select a node with deficit if there is one. Otherwise, select any node with flow leaving. 2 4 1 1 6 6 4 3 5 5 4
Carry out a dfs.
Determine the capacity of W.

14. Updates and the final flow decomposition
Add the cycle flow to the decomposition 2 4 1 1 6 6 4
update the current flow 3 5
2 units around
3 units around
2 units in
1 unit around
3 units around
4 units around 5-6-5
cycle flows
path flows

15. 15.082J / 6.855J / ESD.78J Network Optimization
MITOpenCourseWare
15.082J / 6.855J / ESD.78J Network Optimization
Fall 2010
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