EMIS 8374

Arc-Path Formulation

Updated 2 April 2008

Slide 2

Example Max Flow Problem

s t1

2 4

3

(0,2)

(0,6)

5(0,5)

6

(0,5) (0,4)

(0,4)

(0,7)

Node-Arc Formulation

v

v

v

xxxxx

xxxx

xxxxx

6,56,4

5,35,26,5

4,26,4

3,15,3

2,15,24,2

3,12,1

0

0

0

0

st

max

,70,40,50

,40,20,60,50

6,56,45,3

5,24,23,12,1

xxx

xxxx

Slide 3

Arc-Path Formulation: Notation

• Notation:• P denotes the set of paths from s to t

• Ap denotes the set of arcs in path p

• Uij denotes the set of paths that use arc (i, j)

• uij denotes the capacity of arc (i, j)

• Decision variables• fp denotes the amount of flow on path p

• v denotes the amount of flow sent from s to t

Slide 4

Arc-Path Formulation: LP

Slide 5

Ppf

Ajiuf

vf

v

p

Upijp

Ppp

ij

0

),(

s.t.

max

Arc-Path Formulation for Example Problem

• Pst = {1, 2, 3} where

– A1={(1, 2), (2, 4), (4, 6)}

– A2={(1, 2), (2, 5), (5, 6)}

– A3= {(1, 3), (3, 5), (5, 6)}

– U1,2 = {1, 2}, U1,3 = {3}

– U2,4 = {1}, U2,5 = {2}

– U3,5 = {3}, U4,6 = {1}, U5,6 = {2, 3}

Slide 6

Arc-Path Formulation: LP

Slide 70,,

6) (5,7

6) (4,4

5) (3,5

5) (2,4

4) (2,2

3) (1,6

2) (1,5

s.t.

max

321

32

1

3

2

1

3

21

321

fff

ff

f

f

f

f

f

ff

vfff

v

Slide 8

Example Max Flow Solution

s t1

2 4

3

(2,2)

(5,6)

5(5,5)

6

(4,5) (2,4)

(2,4)

(7,7)

f1 = 2

Slide 9

Example Max Flow Solution

s t1

2 4

3

(2,2)

(5,6)

5(5,5)

6

(4,5) (2,4)

(2,4)

(7,7)

f1 = 2 f2 = 2

Slide 10

Example Max Flow Solution

s t1

2 4

3

(2,2)

(5,6)

5(5,5)

6

(4,5) (2,4)

(2,4)

(7,7)

f1 = 2 f2 = 2 f3 = 5

Arc-Path Model: Pros and Cons

• Pros– Facilitates modeling side constraints on paths

(e.g., hop-count limits)– Explicitly maps flows to paths

• Cons– Doesn’t guarantee integer solutions– Requires explicitly listing paths in P

Slide 11