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and 6.855J Cycle Canceling Algorithm

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2 A minimum cost flow problem , $4 20, $1 20, $2 25, $2 25, $5 20, $6 30, $

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

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4 Capacities on the Residual Network

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5 Costs on the Residual Network 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 The capacity of this cycle is 15. Form the next residual network.

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7 Capacities on the residual network

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8 Costs on the residual network Find a negative cost cycle, if there is one. 5

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

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Capacities on the residual network

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11 Costs in the residual network Find a negative cost cycle, if there is one.

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

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13 Capacities on the residual network

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14 Costs in the residual network 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

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16 Capacities on the residual network

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17 Costs in the residual network 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 Let d(j) be the shortest path distance from node 1 to node j. Next let (j) = -d(j) And compute c

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19 Reduced costs in the residual network The reduced costs in G(x*) for the optimal flow x* are all non-negative.

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