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15.082 and 6.855J Cycle Canceling Algorithm

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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

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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.

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

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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.

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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.

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

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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

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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

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

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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.

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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

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

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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.

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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

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

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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?

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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

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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.

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