# maximum flow

Post on 14-Jul-2015

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Maximum Flow

The Standard Maximum Flow Problem

(flow-capacity) ,

X Y . ,

Rephrasing in terms of graph theory

- (directed graph)

(capacity) c X, Y ,

(flow) f

f

Residual networks

. : f c

: f

Augmenting paths

Residual network

(path capacity)

How to Solve It (1/N)

Figure 2b Residual network Augmenting path? X-A-C-Y

C-Y 1

1

How to Solve It (2/N)

Figure 3b Residual network Augmenting path? X-A-C-B-D-E-Y

C-B 1

1

How to Solve It (3/N)

Figure 1b Residual network Augmenting path?

The Ford-Fulkerson method

//Assumption 1: capacities and flows of the edges being integers

//Assumption 2: path-capacity being positive

Begin

x := 0

create the residual network G(x)

while there is some augmenting path from s to t in G(x)

begin

let P be a path from s to t in G(x)

min:= Capacity of P

Increase flow by min along P

Update G(x)

end

End { the flow x is now maximum }

Correctness of the Ford-Fulkerson

method Augmenting path maximum flow?

Maximum flow Augmenting path

Maximum flow

Augmenting path maximum flow.

Choosing Good Augmenting Paths

Ford-Fulkerson algorithm augmenting path maximum flow

augmenting path 1

(shortest path)

(maximum path capacity)

Reduction of multiple-source/

multiple sink

dummy vertex

super-source: .

super-sink: . .

Eliminating vertex-capacities

2,

,

Maximum Bipartite Matching

11 ( X, Y )

super-source: 1. 1

A B

super-sink

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