1. EMIS 8374 Maximum Concurrent Flow Updated 3 April 2008

2. Maximum Concurrent Flow Problem (MCFP)
Input
Undirected graph G = (V, E) with capacity uij for each edge {i, j} in E
Upper Triangular demand matrix D where dij is the demand for flow between vertex i and vertex j
Optimization Problem
Find a feasible flow and throughput value z such that
Each vertex pair (i, j) receives zdij units of flow
The throughput z is maximized

3. Edge-Path Formulation: Notation
Pij denotes the set of paths between i and j
Ep denotes the set of edges in path p
Uij denotes the set of paths that use edge {i, j}
Decision variables
fp denotes the amount of flow on path p
z denotes the value of the concurrent of flow (throughput)

4. Edge-Path Formulation: LP

5. Example Graph G 1 2 4 2 1 1 1 6 1 2 1 3 5

6. MCFP Example 1
Example graph G with demand matrix D = 1 2 3 4 5 6 -

7. Edge-Path Formulation for Example Problem
P16 = {1, 2, 3, 4} where
E1={{1, 2}, {2, 4}, {4, 6}}
E2={{1, 2}, {2, 5}, {5, 6}}
E3= {{1, 3}, {3, 5}, {5, 6}}
E4= {{1, 3}, {3, 5}, {2, 5}, {2, 4}, {4, 6}}
U1,2 = {1, 2}, U1,3 = {3, 4}
U2,4 = {1, 4}, U2,5 = {2, 4}
U3,5 = {3, 4}, U4,6 = {1, 4}, U5,6 = {2, 3}

8. Edge-Path Formulation: LP
Optimal Solution:
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9. MCFP Example 2 Example graph G with demand matrix D = 1 2 3 4 5 6 -
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10. Edge-Path Formulation for MCFP Ex. 2
P16 = {1, 2, 3, 4} where
E1={{1, 2}, {2, 4}, {4, 6}}
E2={{1, 2}, {2, 5}, {5, 6}}
E3= {{1, 3}, {3, 5}, {5, 6}}
E4= {{1, 3}, {3, 5}, {2, 5}, {2, 4}, {4, 6}}
P25 = {5, 6 7} where
E5={{2, 5}}
E6={{1, 2}, {1, 3}, {3, 5}}
E7= {{2, 4}, {4, 6}, {5, 6}}
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11. Edge-Path Formulation for MCFP Ex. 2
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12. Edge-Path LP for MCFP Example 2
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13. Upper Bounds on z 1 2 4 2 1 1 1 6 1 2 3z  3 1 3 5 z  1 d16 = 3
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14. Upper Bounds on z 1 2 4 2 1 1 1 6 1 2 3z + 2z  3 1 3 5 z  0.6
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15. Optimal Solution for MCFP Ex. 2
Each pair gets 60% of its demand
1.8 units between 1 and 6
1.2 units between 2 and 5
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16. Optimal Solution for MCFP Ex. 21 2 4
0.6
0.4 2
0.4 1
0.6 1 1
0.6
0.4 1 6 1 1 2
0.6 1
0.4
0.4 1 3 5
0.6
0.6
0.4
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17. MCFP Example 3: Uniform Case
Example graph G with uij = 1 for all edges and demand matrix D = 1 2 3 4 5 6 -
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18. Upper Bounds on z 2 4 1 6 (1)(5)z  2 3 5 z  0.4 dij = 1 uij = 1
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19. Upper Bounds on z 2 4 1 6 (2)(4)z  3 3 5 z  0.375 dij = 1 uij = 1
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20. Upper Bounds on z 2 4 1 6 (4)(2)z  2 3 5 z  0.25 dij = 1 uij = 1
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21. Optimal Solution for MCFP Ex. 32 4 1 6 3 5
0.25
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22. Optimal Solution for MCFP Ex. 32 4 1 6 3 5
0.25
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23. Optimal Solution for MCFP Ex. 32 4 1 6 3 5
0.25
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24. Optimal Solution for MCFP Ex. 32 4 1 6 3 5
0.25
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25. Optimal Solution for MCFP Ex. 3
(1,1) 2 4
(1,1)
(0.75,1) 1
(0.75,1) 6
(0.75,1)
(1,1) 3 5
(1,1)
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26. Residual Graph for MCFP Ex. 32 4
(0.25,) 1
(0.25) 6
(0.25) 3 5
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27. Example Graph K2,3 2 4 (4)(1)z  2 z  0.5 z* = 3/7 1 3 5 dij = 1
uij = 1 2 4
(4)(1)z  2
z  0.5
z* = 3/7 1 3 5

28. Example Graph K2,3 2 4 (4)(1)z  2 z  0.5 z* = 3/7 1 3 5 dij = 1
uij = 1 2 4
(4)(1)z  2
Direct flow = 3/7
z  0.5
z* = 3/7 1 3 5

29. Example Graph K2,3 2 4 (4)(1)z  2 z  0.5 z* = 3/7 1 3 5 dij = 1
uij = 1 2 4
(4)(1)z  2
Direct flow = 3/7
z  0.5
2-i-4 flow = 1/7
z* = 3/7
odd-even-odd
flow = 3/14 1 3 5

30. Maximum Concurrent Flow Problem (MCFP)
Provides a way of finding a fair flow in a congested network
Generalization of the standard s-t Maximum Flow Problem
The maximum value of the concurrent flow is less than or equal to the density of the sparsest cut where the density of a cut is defined as the capacity of the cut divided by the demand across the cut
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