cyclecancelingalgorithm

# cyclecancelingalgorithm - 15.082 and 6.855J Cycle Canceling...

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1 15.082 and 6.855J Cycle Canceling Algorithm

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2 A minimum cost flow problem 1 2 4 3 5 10, \$4 20, \$1 20, \$2 25, \$2 25, \$5 20, \$6 30, \$7 25 0 0 0 -25
3 The Original Capacities and Feasible Flow 1 2 4 3 5 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.

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4 Capacities on the Residual Network 1 2 4 3 5 10 20 20 5 25 10 10 15 5 20 10
5 Costs on the Residual Network 1 2 4 3 5 2 2 7 -7 -2 -2 -1 -4 5 Find a negative cost cycle, if there is one.

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6 Send flow around the cycle Send flow around the negative cost cycle 1 2 4 3 5 20 25 15 The capacity of this cycle is 15. Form the next residual network.
7 Capacities on the residual network 1 2 4 3 5 10 20 5 20 10 10 25 5 20 10 15

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8 Costs on the residual network 1 2 4 3 5 2 2 6 7 -7 -2 -2 -1 -4 -6 Find a negative cost cycle, if there is one. 5
9 Send flow around the cycle 1 2 4 3 5 Send flow around the negative cost cycle The capacity of this cycle is 10. Form the next residual network. 20 20 10

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10 Capacities on the residual network 1 2 4 3 5 10 10 5 20 10 20 25 15 10 15 10
11 Costs in the residual network 1 2 4 3 5 1 2 6 7 -7 -6 -2 -1 -4 Find a negative cost cycle, if there is one.

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## cyclecancelingalgorithm - 15.082 and 6.855J Cycle Canceling...

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