14_Min_Cost_Flows_2

# 14_Min_Cost_Flows_2 - 15.082 and 6.855J The Successive...

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1 15.082 and 6.855J The Successive Shortest Path Algorithm and the Capacity Scaling Algorithm for the Minimum Cost Flow Problem

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2 Pseudo-Flows A pseudo-flow is a "flow" vector x such that 0 x u. Let e(i) denote the excess (deficit) at node i. The infeasiblity of the pseudo-flow is e(i)>0 e(i).
3 Supplies/Demands, Capacities, and Flows 1 2 3 5 4 10, 5 20, 0 20, 0 25, 13 25, 0 20, 20 30, 3 23 5 -2 -7 -19 What is e(i) for each node i? What is the infeasibility of the flow?

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4 Optimality Conditions Let π be a vector of node potentials. Let x be a pseudo-flow. For each arc (i,j) G(x), the reduced cost of (i,j) is c π ij = c ij - π i + π j . We say that a pseudo-flow x satisfies the optimality conditions (is dual feasible ) if c π ij 0 for each (i,j) G(x). Overview of the algorithm Start with a pseudo-flow x that satisfies the optimality conditions. At each iteration maintain the optimality conditions, but continually reduce the infeasibility. :
5 Maintaining optimality conditions 1 2 3 5 4 10 20 20 25 25 20 30 23 5 -2 -7 Suppose that red arcs have a reduced cost of 0 -19 A dual feasible network with excesses/deficits and capacities Optimality conditions are maintained if flow is sent on a red path.

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6 Sending Flow Along Paths with Reduced Cost of 0 If each arc of P G(x) has a reduced cost of 0, then sending flow along P maintains the optimality conditions. To reduce infeasibility, send flow from an excess node to a deficit node. Next : how to ensure that there is a path whose reduced cost is 0.
Find a shortest path node from node i, and update potentials. 1

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## This note was uploaded on 05/05/2010 for the course EE 15.082 taught by Professor Orlin during the Spring '10 term at Visayas State University.

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14_Min_Cost_Flows_2 - 15.082 and 6.855J The Successive...

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