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The Multicommodity Flow Problem. Updated 21 April 2008. Problem Inputs. LP Formulation. Figure 17.3 from AMO (costs for all k ). 20. 20. 20. 1. 2. 3. 4. 5. 5. 5. 5. 10. 10. 10. 5. 6. 7. 8. 5. 5. 5. 5. 5. 5. 5. 9. 10. 11. 12. 5. 5. 5. 5. 0. 0. 0. - PowerPoint PPT Presentation
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The Multicommodity Flow Problem
Updated 21 April 2008
Problem Inputs
Multicommodity Flows Slide 2
) ,( arc ofcapacity
) ,( arcon commodity ofamount maximum
) ,( arcon commodity for cost unit per
scommoditie ofset
network),(
jiU
jiku
jikc
K
ANG
ij
kij
kij
LP Formulation
Multicommodity Flows Slide 3
AjiUx
AjiKkux
NiKkbxx
xc
Kkij
kij
kij
kij
ki
AijNj
kji
AjiNj
kij
Kk Aji
kij
kij
),(
),(,0
,s.t.
min
),(:),(:
),(
ijUK capacity sharedby linked MCNFPs
Multicommodity Flows
Figure 17.3 from AMO (costs for all k)
1 42 320 20 20
5 86 710 10 10
5 5 5 5
9 1210 115 5 5
13 1614 150 0 0
5 5 5 5
5 5 5 5
Slide 4
Multicommodity Flows
Figure 17.3 from AMO (Uij)
1 42 3 15
5 86 7 15
9 1210 11 15
13 1614 15 15
Slide 5
Multicommodity Flows
Figure 17.13 from AMO (Commodities)
Commodity Source Sink Units
1 1 4 10
2 5 8 10
3 9 12 10
4 13 16 10
Slide 6
Multicommodity Flows
Routing for Commodities 1, 2, and 4
1 42 3
5 86 7
9 1210 11
13 1614 15
10
1010 10 10
10
1010 10 10
10 10 10
Slide 7
Multicommodity Flows
Routing for Commodity 3
1 42 3
5 86 7
9 1210 11
13 1614 15
5 5 5
5 5 55 5
Slide 8
Multicommodity Flows
Total Flow
1 42 3
5 86 7
9 1210 11
13 1614 15
10
1015 15 15
10
1010 10 10
15 15 155 5
Slide 9
Multicommodity Flows
Example 2
2
1 3
)1,0,0,0(),,,( 321 ijijijij Uccc
),1,1,1(),,,( 321 ijijijij Uccc
k s t b
1 3 2 1
2 1 3 1
3 2 1 1
Slide 10
Multicommodity Flows
Example 2: Routing for Commodity 1
2
1 30.50.5
0.5Cost = 0.5
k s t b
1 3 2 1
2 1 3 1
3 2 1 1
)1,0,0,0(),,,( 321 ijijijij Uccc
),1,1,1(),,,( 321 ijijijij Uccc
Slide 11
Multicommodity Flows
Example 2: Routing for Commodity 2
2
1 3
0.5 0.5
0.5
Cost = 0.5
k s t b
1 3 2 1
2 1 3 1
3 2 1 1)1,0,0,0(),,,( 321 ijijijij Uccc
),1,1,1(),,,( 321 ijijijij Uccc
Slide 12
Multicommodity Flows
Example 2: Routing for Commodity 3
2
1 3
0.50.5
0.5 Cost = 0.5
)1,0,0,0(),,,( 321 ijijijij Uccc
),1,1,1(),,,( 321 ijijijij Uccc
k s t b
1 3 2 1
2 1 3 1
3 2 1 1
Slide 13
Multicommodity Flows
Example 2: Total Flow
2
1 3
1
1
0.5 Cost = 1.5
1
0.5
0.5
)1,0,0,0(),,,( 321 ijijijij Uccc
),1,1,1(),,,( 321 ijijijij Uccc
k s t b
1 3 2 1
2 1 3 1
3 2 1 1
Slide 14
Multicommodity Flows
Example 2: Optimal Integral Flow
2
1 3
Cost = 21 (k =1)
1 (k = 2)
1 (k = 3)1 (k = 3)
)1,0,0,0(),,,( 321 ijijijij Uccc
),1,1,1(),,,( 321 ijijijij Uccc
k s t b
1 3 2 1
2 1 3 1
3 2 1 1
Slide 15
Multicommodity Flows
Complexity
• The bundling constraints make the multicommodity flow problem with integral flows significantly more difficult to solve than pure network flow problems.
• This problem belongs to the class of theoretically intractable NP-hard optimization problems.
Slide 16
Multicommodity Flows
NP-hard Problems
• Multicommodity Flow belongs to the class of NP-hard problems for which no known polynomial time algorithms exist.
• Other NP-hard problems: TSP, network design, longest path, knapsack, integer programming.
• If there exists a polynomial time algorithm for any NP-hard problem, then there is one for every NP-hard problem.
• Whether or not such an algorithm exists is a fundamental unsolved problem in theoretical computer science and OR.
Slide 17
