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 Outline Problem 1 Model 2 Worked Example 3 Results 4 Helmut Simonis Hybrid Model for RWA 3
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
Problem Model Worked Example Results Outline Problem 1 Model 2 Worked Example 3 Results 4 Helmut Simonis Hybrid Model for RWA 5
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
Problem Model Worked Example Results Example Network (NSF , 5 wavelengths) 6 10 3 14 2 5 7 4 8 9 13 1 11 12 Helmut Simonis Hybrid Model for RWA 7
Problem Model Worked Example Results Lightpath from node 5 to node 13 (5 ⇒ 13) 6 10 3 14 2 5 7 4 8 9 13 1 11 12 Helmut Simonis Hybrid Model for RWA 8
Problem Model Worked Example Results Conflict with demand 1 ⇒ 12: Use different frequencies 6 10 3 14 5 7 2 4 8 9 13 1 11 12 Helmut Simonis Hybrid Model for RWA 9
Problem Model Worked Example Results Conflict with demand 1 ⇒ 12: Use different path 6 10 3 14 2 5 7 4 8 9 13 1 11 12 Helmut Simonis Hybrid Model for RWA 10
Problem Model Worked Example Results Conflict with demand 1 ⇒ 12: Reject demand 6 10 3 14 2 5 7 4 8 9 13 1 11 12 Helmut Simonis Hybrid Model for RWA 11
Problem Model Worked Example Results Outline Problem 1 Model 2 Worked Example 3 Results 4 Helmut Simonis Hybrid Model for RWA 12
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 Helmut Simonis Hybrid Model for RWA 13
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 Helmut Simonis Hybrid Model for RWA 14
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 Helmut Simonis Hybrid Model for RWA 15
Problem Model Worked Example Results Notation y sd , integer number of accepted demands from s to d z se , integer capacity used on edge e to satisfy demands sourced in s C , number of available wavelengths, edge capacity P sd , requested number of demands from s to d T s , total number of requested demands sourced from s D s , nodes which have a requested demand sourced in s Helmut Simonis Hybrid Model for RWA 16
Problem Model Worked Example Results Model (Step 1) � � max y sd s ∈ N d ∈ D s s.t. y sd ∈ { 0 , 1 ... P sd } , z se ∈ { 0 , 1 ... T s } � ∀ e ∈ E : z se ≤ C s ∈ N � ∀ s ∈ N : z se = 0 e ∈ In ( s ) � � ∀ s ∈ N , ∀ d ∈ D s : z se = z se + y sd e ∈ In ( d ) e ∈ Out ( d ) � � ∀ s ∈ N , ∀ n � = s , n / ∈ D s : z se = z se e ∈ In ( n ) e ∈ Out ( n ) Helmut Simonis Hybrid Model for RWA 17
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 Helmut Simonis Hybrid Model for RWA 18
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 Helmut Simonis Hybrid Model for RWA 19
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 Helmut Simonis Hybrid Model for RWA 20
Problem Model Worked Example Results Model (Step 2) X d 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 Helmut Simonis Hybrid Model for RWA 21
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 Helmut Simonis Hybrid Model for RWA 22
Problem Model Worked Example Results Solution Approach MIP Resource Model Extract Accepted Demands FD Graph Coloring Remove Demand Yes Provide Explanation Infeasible No Solution Helmut Simonis Hybrid Model for RWA 23
Problem Model Worked Example Results Outline Problem 1 Model 2 Worked Example 3 Results 4 Helmut Simonis Hybrid Model for RWA 24
Problem Model Worked Example Results Demand Matrix (100 Demands) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 1 1 1 1 1 1 1 Color Distance 1 1 1 2 1 2 2 2 1 1 1 1 1 2 3 3 1 3 1 1 1 4 4 1 1 1 1 5 ≥ 5 2 2 1 1 1 1 6 1 1 1 1 1 2 7 1 1 8 1 1 1 1 1 9 2 1 1 1 1 1 1 2 10 1 1 1 1 2 1 2 1 11 2 2 1 2 1 2 2 12 1 2 1 1 13 1 1 1 1 3 1 1 14 Helmut Simonis Hybrid Model for RWA 25
Problem Model Worked Example Results Source Model Solution Color Type Source Sink Source Node 1 Unreached 1 Chosen Link 1 1 1 1 1 1 2 2 1 1 1 3 S Helmut Simonis Hybrid Model for RWA 26
Problem Model Worked Example Results Source Model Solution Color Type Source Sink Source Node 2 Unreached Chosen Link 1 1 S 2 1 1 2 1 2 2 Helmut Simonis Hybrid Model for RWA 27
Problem Model Worked Example Results Source Model Solution Color Type Source Sink Source Node 3 Unreached 2 1 Chosen Link 4 1 S 1 1 3 1/2 3 1 1 1 1 1 2 1 1 1 Helmut Simonis Hybrid Model for RWA 28
Problem Model Worked Example Results Source Model Solution Color Type Source Sink Source Node 4 Unreached Chosen Link 1 1 1 3 1 4 1 S 1 1 1 1 1 Helmut Simonis Hybrid Model for RWA 29
Problem Model Worked Example Results Source Model Solution Color Type Source Sink Source Node 5 Unreached 1 1 Chosen Link 1 1 3 1 1 S 1 1 1 1 1 Helmut Simonis Hybrid Model for RWA 30
Problem Model Worked Example Results Source Model Solution Color Type Source Sink Source Node 6 Unreached 1 S Chosen Link 1 2 5 1 2 1 2 2 2 1 1 1 Helmut Simonis Hybrid Model for RWA 31
Problem Model Worked Example Results Source Model Solution Color Type Source Sink Source Node 7 Unreached 1 1 Chosen Link 1 2 S 3 4 1 3 1 1 1 2 1 1 2 Helmut Simonis Hybrid Model for RWA 32
Problem Model Worked Example Results Source Model Solution Color Type Source Sink Source Node 8 Unreached Chosen Link 2 1 S 1 1 1 1 Helmut Simonis Hybrid Model for RWA 33
Problem Model Worked Example Results Source Model Solution Color Type Source Sink Source Node 9 Unreached Chosen Link 1 1 1 1 1 1 1 S 1 1 1 1 1 1 1 Helmut Simonis Hybrid Model for RWA 34
Problem Model Worked Example Results Source Model Solution Color Type Source Sink Source Node 10 Unreached 1 Chosen Link S 4 1 2 2 4 2 1 1 2 1 1 1 1 1 1 Helmut Simonis Hybrid Model for RWA 35
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