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EMIS 8374 Maximum Concurrent Flow Updated 3 April 2008

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... 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 ... – PowerPoint PPT presentation

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Title: EMIS 8374 Maximum Concurrent Flow Updated 3 April 2008


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
  • 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
1 - 0 0 0 0 1
2 - - 0 0 0 0
3 - - - 0 0 0
4 - - - - 0 0
5 - - - - - 0
6 - - - - - -
7
Edge-Path Formulation for Example Problem
  • P16 1, 2, 3, 4 where
  • E11, 2, 2, 4, 4, 6
  • E21, 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
Slide 8
9
MCFP Example 2
  • Example graph G with demand matrix D

1 2 3 4 5 6
1 - 0 0 0 0 3
2 - - 0 0 2 0
3 - - - 0 0 0
4 - - - - 0 0
5 - - - - - 0
6 - - - - - -
Slide 9
10
Edge-Path Formulation for MCFP Ex. 2
  • P16 1, 2, 3, 4 where
  • E11, 2, 2, 4, 4, 6
  • E21, 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
  • E52, 5
  • E61, 2, 1, 3, 3, 5
  • E7 2, 4, 4, 6, 5, 6

Slide 10
11
Edge-Path Formulation for MCFP Ex. 2
  • U1,2 1, 2, 6
  • U1,3 3, 4, 6
  • U2,4 1, 4, 7
  • U2,5 2, 4, 5
  • U3,5 3, 4, 6
  • U4,6 1, 4, 7
  • U5,6 2, 3, 7

Slide 11
12
Edge-Path LP for MCFP Example 2
Slide 12
13
Upper Bounds on z
1
d16 3 d25 2
2
4
2
1
1
1
6
1
2
3z ? 3
1
3
5
z ? 1
Slide 13
14
Upper Bounds on z
1
d16 3 d25 2
2
4
2
1
1
1
6
1
2
3z 2z ? 3
1
3
5
z ? 0.6
Slide 14
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

Slide 15
16
Optimal Solution for MCFP Ex. 2
1
2
4
0.6
0.4
2
1
0.4
0.6
1
1
0.6
0.4
1
6
1
1
2
1
0.6
0.4
0.4
1
3
5
0.6
0.4
0.6
Slide 16
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
1 - 1 1 1 1 1
2 - - 1 1 1 1
3 - - - 1 1 1
4 - - - - 1 1
5 - - - - - 1
6 - - - - - -
Slide 17
18
Upper Bounds on z
dij 1 uij 1
2
4
1
6
(1)(5)z ? 2
3
5
z ? 0.4
Slide 18
19
Upper Bounds on z
dij 1 uij 1
2
4
1
6
(2)(4)z ? 3
3
5
z ? 0.375
Slide 19
20
Upper Bounds on z
dij 1 uij 1
2
4
1
6
(4)(2)z ? 2
3
5
z ? 0.25
Slide 20
21
Optimal Solution for MCFP Ex. 3
2
4
1
6
3
5
Slide 21
22
Optimal Solution for MCFP Ex. 3
2
4
1
6
3
5
Slide 22
23
Optimal Solution for MCFP Ex. 3
2
4
1
6
3
5
Slide 23
24
Optimal Solution for MCFP Ex. 3
2
4
1
6
3
5
Slide 24
25
Optimal Solution for MCFP Ex. 3
(1,1)
2
4
(1,1)
(0.75,1)
1
6
(0.75,1)
(0.75,1)
(1,1)
3
5
(1,1)
Slide 25
26
Residual Graph for MCFP Ex. 3
2
4
(0.25,)
1
6
(0.25)
(0.25)
3
5
Slide 26
27
Example Graph K2,3
dij 1 uij 1
2
4
(4)(1)z ? 2
z ? 0.5
z 3/7
3
1
5
28
Example Graph K2,3
dij 1 uij 1
2
4
(4)(1)z ? 2
Direct flow 3/7
z ? 0.5
z 3/7
3
1
5
29
Example Graph K2,3
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
3
1
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

Slide 30
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