[PPT] - Chapter 18: A Hybrid Model for the Routing and Wavelength Assignment PowerPoint Presentation

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Problem Model Worked Example Results

Chapter 18: A Hybrid Model for the Routing and Wavelength Assignment Problem

Helmut Simonis

Cork Constraint Computation Centre Computer Science Department University College Cork Ireland

ECLiPSe ELearning

Overview Helmut Simonis Hybrid Model for RWA 1

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Problem Model Worked Example Results

Licence

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Problem Model Worked Example Results

Outline

1

Problem

2

Model

3

Worked Example

4

Results

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Problem Model Worked Example Results

What We Want to Introduce

Hybridisation by decomposition Combination of MIP and FD solver Best current solution to routing and wavelength assignment problem

Helmut Simonis Hybrid Model for RWA 4

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Problem Model Worked Example Results

Outline

1

Problem

2

Model

3

Worked Example

4

Results

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Problem Model Worked Example Results

Problem Definition

Routing and Wavelength Assignment (Demand Acceptance) In an optical network, traffic demands between nodes are assigned to a route through the network and a specific

wavelength. The route (called lightpath) must be a simple path

from source to destination. Demands which are routed over the same link must be allocated to different wavelengths, but wavelengths may be reused for demands which do not meet. The objective is to find a combined routing and wavelength assignment which maximizes the number of accepted demands.

Helmut Simonis Hybrid Model for RWA 6

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Problem Model Worked Example Results

Example Network (NSF , 5 wavelengths)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

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Problem Model Worked Example Results

Lightpath from node 5 to node 13 (5 ⇒ 13)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

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Problem Model Worked Example Results

Conflict with demand 1 ⇒ 12: Use different frequencies

1 2 3 4 5 6 7 8 9 10 11 12 13 14

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Problem Model Worked Example Results

Conflict with demand 1 ⇒ 12: Use different path

1 2 3 4 5 6 7 8 9 10 11 12 13 14

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Problem Model Worked Example Results

Conflict with demand 1 ⇒ 12: Reject demand

1 2 3 4 5 6 7 8 9 10 11 12 13 14

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Problem Model Worked Example Results

Outline

1

Problem

2

Model

3

Worked Example

4

Results

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Problem Model Worked Example Results

Solution Approaches

Greedy heuristic Optimization algorithm for complete problem Decomposition into two problems

Route maximal number of demands Assign wavelengths

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Problem Model Worked Example Results

Solution Approaches

Greedy heuristic Optimization algorithm for complete problem Decomposition into two problems

Route maximal number of demands Assign wavelengths

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Problem Model Worked Example Results

Step 1: Route Maximal Number of Demands

Ignore wavelengths Capacity constraints on all links Solve as MIP problem Source aggregation Find DAG to supply (all) demands with shared source Maximize number of accepted demands

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Problem Model Worked Example Results

Notation

ysd, integer number of accepted demands from s to d zse, integer capacity used on edge e to satisfy demands sourced in s C, number of available wavelengths, edge capacity Psd, requested number of demands from s to d Ts, total number of requested demands sourced from s Ds, nodes which have a requested demand sourced in s

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Problem Model Worked Example Results

Model (Step 1)

max

s∈N
d∈Ds

ysd s.t.

ysd ∈ {0, 1...Psd}, zse ∈ {0, 1...Ts} ∀e ∈ E :

s∈N

zse ≤ C ∀s ∈ N :

e∈In(s)

zse = 0 ∀s ∈ N, ∀d ∈ Ds :

e∈In(d)

zse =

e∈Out(d)

zse + ysd ∀s ∈ N, ∀n = s, n / ∈ Ds :

e∈In(n)

zse =

e∈Out(n)

zse

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Problem Model Worked Example Results

Observation

Optimal cost is upper bound for full problem LP Relaxation is also upper bound for full problem No 0/1 variables in model Source aggregation has massive impact on efficiency

Much better than treating each demand on its own Reason 1: Reduced number of variables Reason 2: Avoids symmetries due to multiple demands between nodes

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Problem Model Worked Example Results

Finding Accepted Demands

Solution to MIP does not tell how demands are routed Program required to convert source “tree” into sets of paths Conversion not deterministic, may allow different solutions Solution may contain loops, these need to be removed

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Problem Model Worked Example Results

Step 2: Assign Wavelengths

For each accepted accepted demand, find frequency All demands routed over a link compete for frequencies Graph coloring problem Graph given as sets of cliques Solve with finite domains If solution found, then optimal for complete problem

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Problem Model Worked Example Results

Model (Step 2)

Xd finite domain variable 1..C for each accepted demand One alldifferent constraint for each edge Many alldifferent constraints are at capacity Possible to improve model

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Problem Model Worked Example Results

What Happens If No Solution Found

Problem infeasible

Remove some demand and try again until solution found Possibly sub-optimal solution of high quality Different solution to MIP problem may lead to optimal solution

No solution found within time limit

Try harder! Improve reasoning and/or search technique Special techniques to show infeasibility

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Problem Model Worked Example Results

Solution Approach

MIP Resource Model Extract Accepted Demands FD Graph Coloring Infeasible Solution Provide Explanation Remove Demand No Yes

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Problem Model Worked Example Results

Outline

1

Problem

2

Model

3

Worked Example

4

Results

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Problem Model Worked Example Results

Demand Matrix (100 Demands)

Color Distance 1 2 3 4 ≥ 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 3 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 2 1 2 1 2 2 1 2 1 2 2 1 2 1 1 1 1 1 1 3 1 1

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Problem Model Worked Example Results

Source Model Solution

Color Type Source Sink Unreached Chosen Link

Source Node 1

S 1 1 1 1 1 1 1

1 2 3 1 1 2

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Problem Model Worked Example Results

Source Model Solution

Color Type Source Sink Unreached Chosen Link

Source Node 2

1 S 1 1 2

1 1 2 2 2

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Problem Model Worked Example Results

Source Model Solution

Color Type Source Sink Unreached Chosen Link

Source Node 3

S 1/2 2 1 1 1 1 1

1 1 3 3 4 1 2 1 1 1 1

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Problem Model Worked Example Results

Source Model Solution

Color Type Source Sink Unreached Chosen Link

Source Node 4

1 S 3 1 1 1

1 1 4 1 1 1 1

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Problem Model Worked Example Results

Source Model Solution

Color Type Source Sink Unreached Chosen Link

Source Node 5

1 S 1 1 1

1 1 3 1 1 1 1 1

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Problem Model Worked Example Results

Source Model Solution

Color Type Source Sink Unreached Chosen Link

Source Node 6

2 2 S 1 1 1 1

2 5 1 1 2 1 2

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Problem Model Worked Example Results

Source Model Solution

Color Type Source Sink Unreached Chosen Link

Source Node 7

1 1 S 1 1 1 2

1 2 3 1 4 3 2 1 1

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Problem Model Worked Example Results

Source Model Solution

Color Type Source Sink Unreached Chosen Link

Source Node 8

S 1 1

2 1 1 1

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Problem Model Worked Example Results

Source Model Solution

Color Type Source Sink Unreached Chosen Link

Source Node 9

1 1 1 S 1 1

1 1 1 1 1 1 1 1 1

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Problem Model Worked Example Results

Source Model Solution

Color Type Source Sink Unreached Chosen Link

Source Node 10

2 1 1 1 1 S 1 1 2

2 4 1 4 1 2 1 1

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Problem Model Worked Example Results

Source Model Solution

Color Type Source Sink Unreached Chosen Link

Source Node 11

1 1 1 1 2 1 2 S 1

2 1 1 3 4 2 2 3

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Problem Model Worked Example Results

Source Model Solution

Color Type Source Sink Unreached Chosen Link

Source Node 12

1/2 2 1 2 1 S 2 2

1 1 2 1 4 4 2 1 5 1

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Problem Model Worked Example Results

Source Model Solution

Color Type Source Sink Unreached Chosen Link

Source Node 13

1 2 1 1 S

1 1 1 2 1 1

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Problem Model Worked Example Results

Source Model Solution

Color Type Source Sink Unreached Chosen Link

Source Node 14

1 1 1 1 3 1 1 S

1 2 3 1 2 1 1 4

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Problem Model Worked Example Results

Accepted Demands (86 Demands)

Color Distance 1 2 3 4 ≥ 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 1 1 1 1 1

0/1

1 1

0/1

2

1/2 2

1 1 1 1 1 1 3 1

0/1 1

1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1

0/1

1 1 2 1

0/1 0/1 1

1 1 2 1 1 1

0/1 0/2 1

2 1

1/2

2 1 2 1 2 2 1

0/2

1 1 1 1

0/1 1

3 1 1

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Problem Model Worked Example Results

Comparison

Demand Matrix Accepted Demands

1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 3 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 2 1 2 1 2 2 1 2 1 2 2 1 2 1 1 1 1 1 1 3 1 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 1 1 1 1 1

0/1

1 1

0/1

2

1/2 2

1 1 1 1 1 1 3 1

0/1 1

1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1

0/1

1 1 2 1

0/1 0/1 1

1 1 2 1 1 1

0/1 0/2 1

2 1

1/2

2 1 2 1 2 2 1

0/2

1 1 1 1

0/1 1

3 1 1

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Problem Model Worked Example Results

Observations

Accepted demands do not always use shortest path Tendency to reject demands with larger minimal distance These use more resources Not compensated in objective function Not fair

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Problem Model Worked Example Results

Resource Requirements

Color Off Capacity 1 2 3 ≥ 4 unused

1 2 3 4 5 6 7 8 9 10 11 12 13 14

3 4 2 2 3 5 3 4 5 5 5 5 3 5 5 5 5 5 4 3 5 4 5 5 3 5 5 5 5 3 5 5 5 5 5 3 5 5 5 5 5 2

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Problem Model Worked Example Results

Graph Coloring Problem

1 2 30 37 39 44 46 82 92 16 71 94 48 86 64 65 78 73 27 28 3 6 40 59 100 20 96 66 84 5 38 51 69 79 91 52 87 57 70 61 72 95 7 14 23 32 42 45 62 63 49 55 8 67 75 97 99 9 36 58 85 90 10 15 17 25 68 11 13 18 43 80 81 88 93 21 26 50 53 22 47 76 74 31 33 35

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Problem Model Worked Example Results

Graph Coloring Solution

Color Wavelength 1 2 3 4 5

30 82 86 37 48 90 25 51 45 91 68 52 94 22 26 61 72 47 67 76 53 81 88 99 93 7 14 23 32 78 62 63 42 21 58 50 85 10 17 15 64 65 80 71 28 70 87 57 95 55 49 96 100 79 13 43 39 44 18 6 16 20 59 27 66 40 69 46 92 1 2 9 36 73 5 38 31 35 33 74 75 97 84 11 8 3

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Problem Model Worked Example Results

Observation

All demands could be assigned to frequencies Optimal solution to complete problem

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Problem Model Worked Example Results

Explaining Infeasibility

109 111 126 129 131 30 71 35 55 1 40 45 78 61 69 106 116 58 128 130 3 60 86 122 135 139 4 27 105 117 121 125 137 138 127 5 6 43 46 48 141 25 120 147 91 114 133 150 9 13 14 22 23 31 102 113 143 148 57 119 149 38 49 87 15 79 98 112 16 20 42 66 18 32 19 28 93 26 146 29 33 41 34 37 59 76 96 83 90 142 99 103

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Problem Model Worked Example Results

Explanations

Ad-hoc: Find pattern which show infeasibility

Find large cliques If clique is larger than number of colors, problem is infeasible This is simple for graphs given

General explanation techniques

Active research area

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Problem Model Worked Example Results

Explanation Method Used: QuickXPlain

Find minimal subset of constraints which is infeasible Conflict set Works when overall problem fails without search Requires some trick to be applied here

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Problem Model Worked Example Results

Assigned Wavelengths

Color Length 1 2 3 4 ≥ 5 unused

Wavelength 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14

26 26 91 30 30 43 23 40 35 75 30 43 35 96 96 55 96 23 55 91 61 26 61 16 64 58 61 64 16 57 43 30 64 58 46 46

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Problem Model Worked Example Results

Assigned Wavelengths

Color Length 1 2 3 4 ≥ 5 unused

Wavelength 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14

86 22 7 59 22 51 48 86 93 7 59 97 86 93 33 20 51 22 49 20 7 36 51 22 85 94 28 65 85 28 65 94 28 93 1 65 36 73 1

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Problem Model Worked Example Results

Assigned Wavelengths

Color Length 1 2 3 4 ≥ 5 unused

Wavelength 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14

25 90 90 25 99 25 99 14 3 69 99 25 13 74 79 69 14 32 79 76 90 70 71 14 90 70 13 71 70 37 13 79 32 92 71

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Problem Model Worked Example Results

Assigned Wavelengths

Color Length 1 2 3 4 ≥ 5 unused

Wavelength 4

1 2 3 4 5 6 7 8 9 10 11 12 13 14

68 45 45 88 17 17 88 63 100 8 15 88 47 38 47 63 66 78 100 21 82 21 82 27 21 95 39 82 18 82 18 9 78 39 9

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Problem Model Worked Example Results

Assigned Wavelengths

Color Length 1 2 3 4 ≥ 5 unused

Wavelength 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14

53 84 52 81 10 10 81 62 6 31 67 10 81 31 5 5 67 62 6 42 87 72 50 72 6 50 72 44 2 87 81 2 80 87 11 44

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Problem Model Worked Example Results

Accepted Demands (86 Demands)

Color Distance 1 2 3 4 ≥ 5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 1 1 1 1 1

0/1

1 1

0/1

2

1/2 2

1 1 1 1 1 1 3 1

0/1 1

1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1

0/1

1 1 2 1

0/1 0/1 1

1 1 2 1 1 1

0/1 0/2 1

2 1

1/2

2 1 2 1 2 2 1

0/2

1 1 1 1

0/1 1

3 1 1

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Problem Model Worked Example Results

Outline

1

Problem

2

Model

3

Worked Example

4

Results

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Problem Model Worked Example Results

Benchmarks

Fixed network structure nsf 14 nodes, 42 edges eon 20 nodes, 78 edges mci 19 nodes, 64 edges brezil 27 nodes, 140 edges Random network structure

Sizes from 30 to 100 nodes Edge density 0.25 500 demands, 30 wavelengths

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Problem Model Worked Example Results

Overall Distribution of Solutions

Type Technique Count Infeasible clique 50 preassign 38 Feasible credit total 59962

f that, credit a units

58861

f that, credit a2 units

940

f that, credit a3 units

161 complete search, BC alldifferent 25 complete search, GAC alldifferent 12

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Problem Model Worked Example Results

Selected Examples (100 Runs Each)

Network Dem. λ Opt. Avg LP Avg MIP Avg FD Max Gap Avg Time Max Time brezil 500 15 98 483.86 483.86 483.84 1.00 0.92 1.34 brezil 600 20 100 590.96 590.96 590.96 0.00 1.00 1.34 brezil 700 20 100 672.53 672.53 672.53 0.00 1.19 1.78 brezil 800 25 99 781.39 781.39 781.37 2.00 1.44 11.47 eon 500 20 100 471.56 471.56 471.56 0.00 0.65 0.77 eon 600 25 100 574.80 574.80 574.80 0.00 0.82 1.13 eon 700 30 100 677.35 677.35 677.35 0.00 1.05 1.81 eon 800 35 100 779.17 779.17 779.17 0.00 1.28 1.94 mci 500 25 100 486.38 486.38 486.38 0.00 0.80 2.28 mci 600 30 100 585.18 585.18 585.18 0.00 1.27 29.81 mci 700 35 100 684.00 684.00 684.00 0.00 1.30 3.53 mci 800 40 100 782.86 782.86 782.86 0.00 1.68 5.21 nsf 500 35 100 495.20 495.20 495.20 0.00 0.50 0.60 nsf 600 40 100 588.63 588.63 588.63 0.00 0.66 0.98 nsf 700 45 100 678.44 678.44 678.44 0.00 0.86 1.35 nsf 800 45 100 727.15 727.15 727.15 0.00 0.95 1.56

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Problem Model Worked Example Results

Compared to MIP Model for Complete Problem

Hybrid Model Full MIP Network Dem. λ Opt. Avg FD Avg Time Max Time Avg Opt Avg Time Max Time brezil 500 15 98 483.84 0.92 1.34 483.86 1218.40 14103.84 brezil 600 20 100 590.96 1.00 1.34 590.96 6076.81 87767.95 brezil 700 25 98 695.48 1.01 1.80 695.48 13623.15 78463.89 brezil 800 25 99 781.37 1.44 11.47 781.39 7567.68 15456.50 eon 500 20 100 471.56 0.65 0.77 471.56 352.21 585.45 eon 600 25 100 574.80 0.82 1.13 574.80 1411.67 2877.88 eon 700 30 100 677.35 1.05 1.81 677.35 1727.52 3568.13 eon 800 35 100 779.17 1.28 1.94 779.17 2485.64 4116.11 mci 500 25 100 486.38 0.80 2.28 486.38 1023.16 1664.31 mci 600 30 100 585.18 1.27 29.81 585.18 1621.30 2895.88 mci 700 35 100 684.00 1.30 3.53 684.00 1987.23 3428.41 mci 800 40 100 782.86 1.68 5.21 782.86 2316.88 4402.44 nsf 500 35 100 495.20 0.50 0.60 495.20 82.85 173.19 nsf 600 40 100 588.63 0.66 0.98 588.63 155.90 373.63 nsf 700 45 100 678.44 0.86 1.35 678.44 205.82 586.61 nsf 800 45 100 727.15 0.95 1.56 727.15 173.53 410.97 Helmut Simonis Hybrid Model for RWA 60

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Problem Model Worked Example Results

Increasing Number of Demands

Network Dem. λ Opt. Avg LP Avg MIP Avg FD Max Gap Avg MIP Time Max MIP Time Avg FD Time Max FD Time eon 800 30 100 741.78 741.78 741.78 0.00 0.15 0.17 0.83 1.61 eon 900 40 100 880.59 880.59 880.59 0.00 0.14 0.16 1.18 2.17 eon 1000 40 100 950.36 950.36 950.36 0.00 0.15 0.17 1.37 3.42 eon 1100 50 100 1082.61 1082.61 1082.61 0.00 0.14 0.16 1.71 2.83 eon 1200 50 100 1156.38 1156.38 1156.38 0.00 0.15 0.17 2.07 5.92 eon 1300 50 100 1219.82 1219.82 1219.82 0.00 0.16 0.17 2.22 5.24 eon 1400 60 100 1361.47 1361.47 1361.47 0.00 0.15 0.16 2.92 4.94 eon 1500 60 99 1428.78 1428.78 1428.77 1.00 0.15 0.17 4.22 106.97 eon 1600 70 100 1565.90 1565.90 1565.90 0.00 0.15 0.16 3.89 8.48 eon 1700 70 100 1637.47 1637.47 1637.47 0.00 0.16 0.17 4.58 13.59 eon 1800 80 100 1769.86 1769.86 1769.86 0.00 0.15 0.16 5.19 8.81 eon 1900 80 99 1844.46 1844.46 1844.45 1.00 0.15 0.17 7.23 163.41 eon 2000 90 100 1972.66 1972.66 1972.66 0.00 0.15 0.17 6.34 9.61 Helmut Simonis Hybrid Model for RWA 61

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Problem Model Worked Example Results

Random Networks (Edge Density 0.25, 100 Runs Each)

Network Dem. λ Opt. Avg LP Avg MIP Avg FD Avg MIP Time Max MIP Time Avg FD Time Max FD Time r30 500 30 100 391.82 391.82 391.82 0.45 0.55 0.12 0.16 r40 500 30 100 424.58 424.58 424.58 1.07 1.23 0.14 0.17 r50 500 30 100 437.69 437.69 437.69 2.13 2.38 0.09 0.13 r60 500 30 100 447.21 447.21 447.21 3.92 4.34 0.08 0.16 r70 500 30 100 453.41 453.41 453.41 6.78 7.50 0.10 0.17 r80 500 30 100 457.65 457.65 457.65 10.75 11.95 0.10 0.17 r90 500 30 100 464.69 464.69 464.69 16.08 17.45 0.08 0.22 r100 500 30 100 466.67 466.67 466.67 22.74 25.22 0.09 0.25 Helmut Simonis Hybrid Model for RWA 62

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Problem Model Worked Example Results

Observations

MIP and LP relaxation of phase 1 are very good bounds Solved to optimality in most cases Simple decomposition quite effective Good solution even if initial graph coloring infeasible Special structure of graph coloring helps FD model

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Problem Model Worked Example Results

Conclusions

Combination of MIP and FD solver in problem decomposition Each doing what they do best

MIP: optimal solution, select items to include FD: find feasible solution, explain infeasibility

Hybrid model produces very high quality results Proven optimality in over 99.85% of problems tested Near optimal solutions by relaxation Much faster than monolithic MIP solution

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More Information

Rajiv Ramaswami and Kumar N. Sivarajan. Routing and wavelength assignment in all-optical networks. IEEE/ACM Trans. Netw., 3(5):489–500, 1995. Dhritiman Banerjee and Biswanath Mukherjee. A practical approach for routing and wavelength assignment in large wavelength-routed optical networks. IEEE Journal on Selected Areas in Communications, 14(5):903–908, June 1996.

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More Information

Brigitte Jaumard, Christophe Meyer, and Babacar Thiongane. ILP formulations for the routing and wavelength assignment problem: Symmetric systems. In M. Resende and P . Pardalos, editors, Handbook of Optimization in Telecommunications, pages 637–677. Springer, 2006. Brigitte Jaumard, Christophe Meyer, and Babacar Thiongane. Comparison of ILP formulations for the RWA problem. Optical Switching and Networking, 4(3-4):157–172, 2007.

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More Information

Ulrich Junker. Quickxplain: Conflict detection for arbitrary constraint propagation algorithms. In IJCAI’01 Workshop on Modelling and Solving problems with constraints (CONS-1), Seattle, WA, USA, August 2001.

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More Information

Helmut Simonis. Constraint applications in networks. In F . Rossi, P . van Beek, and T. Walsh, editors, Handbook

f Constraint Programming. Elsevier, 2006.

Helmut Simonis. A hybrid constraint model for the routing and wavelength assignment problem. CP 2009, Lisbon, September 2009. http://4c.ucc.ie/~hsimonis/rwa.pdf

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More Information

Helmut Simonis. Solving the static design routing and wavelength assignment problem. CSCLP 2009, Barcelona, Spain, June 2009.

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