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

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

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

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

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

Send flow around the cycle 2 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.

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

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.

Send flow around the cycle 2 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.

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

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.

Send Flow Around the Cycle 10 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.

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

Costs in the residual network 4 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.

Send Flow Around the Cycle 2 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.

Capacities on the residual network 5 2 4 25 5 20 5 1 25 5 15 20 5 3 5 20

Costs in the residual network 4 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?

Compute shortest distances in the residual network 7 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

Reduced costs in the residual network 7 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

15.082J / 6.855J / ESD.78J Network Optimization MITOpenCourseWare http://ocw.mit.edu 15.082J / 6.855J / ESD.78J Network Optimization Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.