08_Max_Flows_1

# 08_Max_Flows_1 - 15.082J and 6.855J March 4, 2003...

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1 15.082J and 6.855J March 4, 2003 Introduction to Maximum Flows

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2 The Max Flow Problem G = (N,A) x ij = flow on arc (i,j) u ij = capacity of flow in arc (i,j) s = source node t = sink node Maximize v Subject to Σ j x ij - Σ k x ki = 0 for each i s,t Σ j x sj = v 0 x ij u ij for all (i,j) A.
3 Maximum Flows We refer to a flow x as maximum if it is feasible and maximizes v. Our objective in the max flow problem is to find a maximum flow. s 1 2 t 10 , 8 8, 7 1, 1 10, 6 6, 5 A max flow problem. Capacities and a non- optimum flow.

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4 The feasibility problem: find a feasible flow retailers 1 2 3 4 5 6 7 8 9 warehouses 6 7 6 5 Is there a way of shipping from the warehouses to the retailers to satisfy demand? 6 5 4 5 4
5 Transformation to a max flow problem 1 2 3 4 5 6 7 8 9 warehouses retailers There is a 1-1 correspondence with flows from s to t with 24 units (why 24?) and feasible flows for the transportation problem. 6 4 5 s 5 t

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6 The feasibility problem: find a matching Is there a way of assigning persons to tasks so that each person is assigned a task, and each task has a person assigned to it? 1 2 3 4 5 6 7 8 persons tasks
7 Transformation to a maximum flow problem Does the maximum flow from s to t have 4 units? 1 2 3 4 5 6 7 8 persons tasks s 1 1 1 1 t 1 1 1 1

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8 The Residual Network s 1 2 t 10 , 8 8, 7 1, 1 10, 6 6, 5 s 1 2 t 2 1 1 4 1 8 5 6 7 i j x u ij ij i j u - x x ij ij ij We let r ij denote the residual capacity of arc (i,j) The Residual Network G(x)
9 A Useful Idea: Augmenting Paths An augmenting path is a path from s to t in the residual network. The residual capacity of the augmenting path P is δ (P) = min{r ij : (i,j) P}. To augment along P is to send d(P) units of flow along each arc of the path. We modify x and the residual capacities appropriately. r ij := r ij - δ (P) and r ji := r ji + δ (P) for (i,j) P. s 1 2 t 2 1 4 8 5 6 7 s 1 2 t 2 1 4 8 6 6 8

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10 The Ford Fulkerson Maximum Flow Algorithm Begin x := 0; create the residual network G(x); while there is some directed path from s to t in G(x) do begin
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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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08_Max_Flows_1 - 15.082J and 6.855J March 4, 2003...

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