Maximum Flow

CSC 172

SPRING 2002

LECTURE 27

Flow Networks

Digraph

Weights, called capacities, on edges

Two distinct veticiesSource, “s” (no incoming edges)

Sink, “t” (no outgoing edges)

Example

Source

s

t

Sink

2

2

2

2

2

3

1

21

1

4

2

Capacity and Flow

Edge Capacities

Non-negative weights on network edges

Flow

function on network edges

0 <= flow <= capacity

flow into vertex == flow out of vertex

Value

combined flow into the sink

Example

Source

s

t

Sink

2

2

2

2

2

3

1

21

1

4

1 21

0 21

0

11

1

1

22

FormallyFlow(u,v) edge(u,v)

Capacity rule edge(u,v)

0 <= flow <= capacity

Conservation rule

uin(v) flow(u,v) = uout(v) flow(v,w)

Value rule

|f| = wout(s) flow(s,w) = uin(t) flow(u,t)

Maximum Flow Problem

Given a network N, find a flow of maximum value

ApplicationsTraffic movement

Hydraulic systems

Electrical circuits

Layout

Example

Source

s

t

Sink

2

2

2

2

2

3

1

21

1

4

2 21

1 11

1

21

0

1

22

Augmenting Path

t

s

2

2

1

12

2

2

12

1

Network with flow 3

Augmenting Path

t

s

2

2

1

12

2

2

12

1

Network with flow 3

t

s

2

2

2

10

2

Augmenting Path

Augmenting Path

t

s

2

2

1

12

2

2

12

1

Network with flow 3

t

s

2

2

2

10

2

Augmenting Path

t

s

2

2

2

12

2

2

02

2Result

Augmenting Path

Forward Edges

flow(u,v) < capacity

we can increase flow

Backward edges

flor(u,v) > 0

flow can be decreased

Max Flow Theorem

A flow has maximum value if and only if it has no augmenting path

Ford & Fulkerson Flow Algorithm

Initialize network with null flow

Method FindFlow

if augmenting path exists then

find augmenting path

increase flow

recursively call FindFlow

Finding Flow

t

s

2

2

0

12

2

0

00

0

Initialize Network

Finding Flow

t

s

2

2

0

12

2

1

11

0

Send one unit of flow through

Finding Flow

t

s

2

2

0

12

2

2

11

1

Send another unit of flow through

Finding Flow

t

s

2

2

1

12

2

2

12

1

Send another unit of flow through

Finding Flow

t

s

2

2

2

12

2

2

02

2

Send another unit of flow through

Note that there is still an augmentingPath that can proceed backwards

Finding Flow

t

s

2

2

2

12

2

2

02

2

Send another unit of flow through

Note that there are no moreAugmenting paths

Residual Network Nf

v

u c(u,v)

f(u,v)

Residual Network Nf

v

u c(u,v)

f(u,v)

v

ucf(u,v)=f(u,v)

cf(u,v)=c(u,v)-f(u,v)

Residual Network

In Nf all edges (w,z) with capacity cf(w,z)=0 are removed

Augmenting Path in N is a direct path in Nf

Augmenting paths can be found by performing a depth-first search on the residual network Nf

Finding Augmenting Path

t

s

2

2

1

12

2

2

12

1

Finding Augmenting Path

t

s

2

2

1

12

2

2

12

1

t

s

1

1

12

2

1

1