1. Cycle Canceling Algorithm
15.082J and 6.855J and ESD.78J
Cycle Canceling Algorithm

2. A minimum cost flow problem
10, $4 2 4
30, $7
25, $5 25 1
20, $2
20, $6
20, $1 3 5
25, $2
-25

3. The Original Capacities and Feasible Flow
10,10 2 4
30,25
25,15 25 1
20,10
20,20
20,0
The feasible flow can be found by solving a max flow. 3 5
25,5
-25

4. Capacities on the Residual Network10 2 4 5 10 25 1 15 10 20 10 20 20 3 5 5

5. Costs on the Residual Network2 4 -4 7 -7 1 2 5 -2 -5 -1 6 2 3 5 -2
Find a negative cost cycle, if there is one.

6. Send flow around the cycle2 4
Send flow around the negative cost cycle 25 1 15 20
The capacity of this cycle is 15. 3 5
Form the next residual network.

7. Capacities on the residual network10 2 4 20 10 10 1 25 20 10 15 5 20 3 5 5

8. Costs on the residual network-4 2 4 7 -7 2 1 5 -2 -6 -1 6 2 3 5 -2
Find a negative cost cycle, if there is one.

9. Send flow around the cycle2 4
Send flow around the negative cost cycle 1 10 20
The capacity of this cycle is 10. 3 5 20
Form the next residual network.

10. Capacities on the residual network10 2 4 20 20 10 1 25 10 10 15 5 10 3 5 15

11. Costs in the residual network-4 2 4 7 -7 1 2 5 1 -1 -6 6 2 3 5 -2
Find a negative cost cycle, if there is one.

12. Send Flow Around the Cycle10 2 4
Send flow around the negative cost cycle 20 10 1 5
The capacity of this cycle is 5. 3 5
Form the next residual network.

13. Capacities on the residual network5 2 4 25 5 15 5 1 25 10 10 20 5 10 3 5 15

14. Costs in the residual network4 2 4 7 -4 -7 1 2 5 -1 1 -2 -6 2 3 5 -2
Find a negative cost cycle, if there is one.

15. Send Flow Around the Cycle2 4
Send flow around the negative cost cycle 1 10 5 10
The capacity of this cycle is 5. 3 5
Form the next residual network.

16. Capacities on the residual network5 2 4 25 5 20 5 1 25 5 15 20 5 3 5 20

17. Costs in the residual network4 2 4 7 -4 -7 1 2 5 -1 1 -6
Find a negative cost cycle, if there is one. 2 3 5 -2
There is no negative cost cycle. But what is the proof?

18. Compute shortest distances in the residual network7 11 4 2 4 7 -4 -7 1 2 5 -1 1 -6
Let d(j) be the shortest path distance from node 1 to node j. 2 3 5 -2 10 12
Next let p(j) = -d(j)
And compute cp

19. Reduced costs in the residual network7 11 2 4 -0 1 2 1 4
The reduced costs in G(x*) for the optimal flow x* are all non-negative. 3 5 10 12

20. 15.082J / 6.855J / ESD.78J Network Optimization
MITOpenCourseWare
15.082J / 6.855J / ESD.78J Network Optimization
Fall 2010
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