1. 15.082 and 6.855J Cycle Canceling Algorithm

2. 2 A minimum cost flow problem 1 24 35 10, $4 20, $1 20, $2 25, $2 25, $5 20, $6 30, $7 25 0 0 0-25

3. 3 The Original Capacities and Feasible Flow 1 24 35 10,10 20,20 20,10 25,5 25,15 20,0 30,25 25 0 0 0-25 The feasible flow can be found by solving a max flow.

4. 4 Capacities on the Residual Network 1 24 35 10 20 5 25 10 15 5 20 10

5. 5 Costs on the Residual Network 1 24 35 2 2 6 7 -7 -5 -2 -4 5 Find a negative cost cycle, if there is one.

6. 6 Send flow around the cycle Send flow around the negative cost cycle 1 24 35 20 25 15 The capacity of this cycle is 15. Form the next residual network.

7. 7 Capacities on the residual network 1 24 35 10 20 5 10 25 5 20 10 15

8. 8 Costs on the residual network 1 24 35 2 2 6 7 -7 -2 -4 -6 Find a negative cost cycle, if there is one. 5

9. 9 Send flow around the cycle 1 24 35 Send flow around the negative cost cycle The capacity of this cycle is 10. Form the next residual network. 20 10

10. Capacities on the residual network 1 24 35 10 5 20 10 20 25 15 10 15 10

11. 11 Costs in the residual network 1 24 35 1 2 2 5 6 7 -7 -6 -2 -4 Find a negative cost cycle, if there is one.

12. 12 Send Flow Around the Cycle 1 24 35 Send flow around the negative cost cycle The capacity of this cycle is 5. Form the next residual network. 10 5 20

13. 13 Capacities on the residual network 1 24 35 5 10 25 5 15 25 15 10 20 10 5 5

14. 14 Costs in the residual network 1 24 35 1 2 2 7 -7 -6 -2 -4 4 -2 5 Find a negative cost cycle, if there is one.

15. 15 Send Flow Around the Cycle Send flow around the negative cost cycle The capacity of this cycle is 5. Form the next residual network. 1 24 35 10 5

16. 16 Capacities on the residual network 1 24 35 5 15 25 5 20 25 20 5 5 5

17. 17 Costs in the residual network 1 24 35 1 2 2 7 -7 -6 -2 -4 4 5 Find a negative cost cycle, if there is one. There is no negative cost cycle. But what is the proof?

18. 18 Compute shortest distances in the residual network 1 24 35 1 2 2 7 -7 -6 -2 -4 4 5 Let d(j) be the shortest path distance from node 1 to node j. Next let (j) = -d(j) 0 7 11 1210 And compute c

19. 19 Reduced costs in the residual network 1 24 35 0 7 11 1210 0 0 2 0 -0 4 0 0 0 0 1 The reduced costs in G(x*) for the optimal flow x* are all non-negative.