Solving the Static Design Routing and Wavelength Assignment Problem
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1 Solving the Static Design Routing and Wavelength Assignment Problem Helmut Simonis Cork Constraint Computation Centre Computer Science Department University College Cork Ireland CSCLP 009, Barcelona Helmut Simonis Static Design RWA
2 Outline Problem Problem Helmut Simonis Static Design RWA
3 Main Points Problem Compare static design and demand acceptance versions of RWA See impact of objective function Compare finite domain, MIP and SAT solutions Helmut Simonis Static Design RWA
4 Outline Problem Problem Helmut Simonis Static Design RWA
5 Problem Definition Routing and Wavelength Assignment (Static Design) 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 minimizes the number of wavelengths used for a given set of demands. Helmut Simonis Static Design RWA
6 RWA Problem Variants Static design Accept all demands Minimize frequencies required Design problem Demand acceptance Number of frequencies fixed Maximize number of demands accepted Operational problem Helmut Simonis Static Design RWA 6
7 Example Network (NSF, nodes) Helmut Simonis Static Design RWA 7
8 Lightpath from node to node ( ) Helmut Simonis Static Design RWA 8
9 Conflict with demand : Use different frequencies Helmut Simonis Static Design RWA 9
10 Conflict with demand : Use different path Helmut Simonis Static Design RWA 0
11 Solution Approaches Greedy heuristic Optimization algorithm for complete problem into two problems Find routing Assign wavelengths Helmut Simonis Static Design RWA
12 Solution Approaches Greedy heuristic Optimization algorithm for complete problem into two problems Find routing Assign wavelengths Helmut Simonis Static Design RWA
13 Outline Problem Problem Helmut Simonis Static Design RWA
14 What is the Objective? Basic model Minimize number of frequencies used on any link Cost of equipment (?) Extended model Minimize overall number of frequencies Cost of renting fibres (?) Helmut Simonis Static Design RWA
15 Notation Problem Network (N,E) directed graph with nodes N and edges E Demands D from source s(d) to sink t(d) Out(n) all links leaving n, In(n) all links entering n Available frequencies Λ 0/ integer variables xde λ, demand d is routed over edge e using frequency λ 0/ integer variables yd λ, demand d is using frequency λ Helmut Simonis Static Design RWA
16 Basic Model s.t. Problem min max e E d D,λ Λ x λ de yd λ {0, }, x de λ {0, } d D : yd λ = λ Λ e E, λ Λ : xde λ d D d D, λ Λ : xde λ = 0, xde λ = y d λ d D, λ Λ : e In(s(d)) e Out(s(d)) xde λ = 0, xde λ = y d λ e Out(t(d)) e In(t(d)) d D, λ Λ, n N \ {s(d), t(d)} : xde λ = xde λ e In(n) e Out(n) Helmut Simonis Static Design RWA 6
17 Extended Model, Additional Variables Minimize the total number of variables used 0/ integer variables z λ, frequency λ is used by at least one demand Helmut Simonis Static Design RWA 7
18 Extended Model min z λ λ Λ s.t. z λ {0, }, yd λ {0, }, x de λ {0, } d D : yd λ = λ Λ d D, e E, λ Λ : xde λ zλ e E, λ Λ : xde λ d D d D, λ Λ : xde λ = 0, xde λ = y d λ d D, λ Λ : e In(s(d)) e Out(s(d)) xde λ = 0, xde λ = y d λ e Out(t(d)) e In(t(d)) d D, λ Λ, n N \ {s(d), t(d)} : xde λ = xde λ e In(n) e Out(n) Helmut Simonis Static Design RWA 8
19 s Problem Scalability Network size Number of demands Symmetries in model Helmut Simonis Static Design RWA 9
20 Improvement: Source Aggregation Combine all demands starting in common source Removes some, but not all symmetries 0/ integer variables x λ se, a demand starting in s is routed over edge e using frequency λ Integer objective z max Integer P sd, number of demands between s and d Set D s, all destinations for demands starting in s Helmut Simonis Static Design RWA 0
21 Source Aggregation, Basic Model min z max s.t. z max {0,... Λ }, xse λ {0, } e E, λ Λ : xse λ s N s N, λ Λ : xse λ = 0 e In(s) s N, d D s, λ Λ : xse λ xse λ e In(d) e Out(d) s N, d D s : xse λ = xse λ + P sd λ Λ e In(d) λ Λ e Out(d) s N, n s, n / D s, λ Λ : xse λ = xse λ e In(n) e Out(n) e E : xse λ zmax s N λ Λ Helmut Simonis Static Design RWA
22 Observations Problem Basic model scales reasonably well Extended model very poor LP relaxation extremely weak LP bound Neither works well enough for larger problem sizes Aggregated model does not directly provide solution Helmut Simonis Static Design RWA
23 Outline Problem Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Problem Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Helmut Simonis Static Design RWA
24 Solution Approach Problem Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model MIP - MIP Based MIP - SAT/FD based decomposition MIP Resource Model Optimization Problem MIP Resource Model Optimization Problem Extract Accepted Demands Extract Accepted Demands MIP Graph Coloring Optimization Problem SAT/FD Graph Coloring Decision Problem Solution Solved? Yes Solution No Increase Nr Wavelengths Helmut Simonis Static Design RWA
25 Idea Problem Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Simplify source aggregation model by ignoring frequencies Integer variables z se, how many demands sourced in s are routed over e Integer objective z max, corresponds to basic problem Constraints independent of number of frequencies, number of demands Helmut Simonis Static Design RWA
26 Phase MIP Problem Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model s.t. min z max z max {0,... Λ }, z se {0,...T s } s N : z se = 0 s N, d D s : s N, n s, n / D s : e E : e In(s) e In(d) e In(n) z se = z se = z se + P sd e Out(d) e Out(n) z se z max s N z se Helmut Simonis Static Design RWA 6
27 Demand Extraction Problem Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Find path for each demand Non-deterministic, backtrack free search Remove loops at same time Procedural Result: Predicate p(d, e) whether demand d is routed over edge e Helmut Simonis Static Design RWA 7
28 Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Resource Requirements (Static Design) Color Off Capacity 0 unused Helmut Simonis Static Design RWA 8
29 Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Compare: Requirements (Demand Acceptance) 6 0 Color Off Capacity 0 unused Helmut Simonis Static Design RWA 9
30 Source Model Solution Source Node Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Color Type Source Sink Unreached Chosen Link S Helmut Simonis Static Design RWA 0
31 Source Model Solution Source Node Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Color Type Source Sink Unreached Chosen Link S Helmut Simonis Static Design RWA
32 Source Model Solution Source Node S Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Color Type Source Sink Unreached Chosen Link Helmut Simonis Static Design RWA
33 Source Model Solution Source Node Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Color Type Source Sink Unreached Chosen Link S 7 Helmut Simonis Static Design RWA
34 Source Model Solution Source Node Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Color Type Source Sink Unreached Chosen Link 8 S Helmut Simonis Static Design RWA
35 Source Model Solution Source Node 6 S Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model 6 Color Type Source Sink Unreached Chosen Link Helmut Simonis Static Design RWA
36 Source Model Solution Source Node 7 Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Color Type Source Sink Unreached Chosen Link S Helmut Simonis Static Design RWA 6
37 Source Model Solution Source Node 8 Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Color Type Source Sink Unreached Chosen Link 8 S Helmut Simonis Static Design RWA 7
38 Source Model Solution Source Node 9 Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Color Type Source Sink Unreached Chosen Link S Helmut Simonis Static Design RWA 8
39 Source Model Solution Source Node 0 6 Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model S 8 Color Type Source Sink Unreached Chosen Link Helmut Simonis Static Design RWA 9
40 Source Model Solution Source Node Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Color Type Source Sink Unreached Chosen Link S Helmut Simonis Static Design RWA 0
41 Source Model Solution Source Node Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Color Type Source Sink Unreached Chosen Link S Helmut Simonis Static Design RWA
42 Source Model Solution Source Node Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Color Type Source Sink Unreached Chosen Link S Helmut Simonis Static Design RWA
43 Source Model Solution Source Node Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model 8 Color Type Source Sink Unreached Chosen Link S Helmut Simonis Static Design RWA
44 Comparison Problem Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Shortest Path Routed Demands Helmut Simonis Static Design RWA
45 Phase MIP Idea Problem Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Assign each demand to a frequency Basic model: Minimize the largest number of frequencies used on any link Extended model: Minimize total number of frequencies 0/ integer variables xd λ, whether demand d uses frequency λ Extended model only: 0/ integer variables z λ Clash constraints: Only one demand can use each frequency on a link Helmut Simonis Static Design RWA
46 Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Phase MIP Model (Basic Problem) min z max s.t. xd λ {0, },z max {0,,..., Λ } d D : xd λ = e E λ Λ : e E : λ Λ {d D p(d,e)} λ Λ {d D p(d,e)} x λ d x λ d z max Helmut Simonis Static Design RWA 6
47 Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Phase MIP Model (Extended Problem) s.t. min z max xd λ {0, }, zλ {0, },z max {0,,..., Λ } d D : xd λ = e E λ Λ : e E : λ Λ {d D p(d,e)} λ Λ {d D p(d,e)} x λ d x λ d z max d D λ Λ : xd λ zλ z λ z max Helmut λ ΛSimonis Static Design RWA 7
48 Finite Domain Model, Idea Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Graph coloring problem Finite domain variables y d, demand d is assigned to frequency y d Two demands routed over same edge must use different frequencies Binary disequality constraints Aggregated to alldifferent constraints Finite domain variables n e, edge e uses n e frequencies nvalue constraint counts number of different values used Helmut Simonis Static Design RWA 8
49 Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Phase Finite Domain Constraints (Basic Model) s.t. min max e E n e y d {0,..., Λ }, n e {0,..., Λ } e E : nvalue(n e, {y d p(d, e)}) e E : alldifferent({y d p(d, e)}) Helmut Simonis Static Design RWA 9
50 Simplification Problem Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model nvalue and alldifferent constraints are over same variable sets Values of alldifferent constraint must be different nvalue constraints can be removed n e variables are not required Not an optimization problem! Helmut Simonis Static Design RWA 0
51 Simplified Basic Model Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model y d {0,..., Λ } e E : alldifferent({y d p(d, e)}) Helmut Simonis Static Design RWA
52 Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Phase Finite Domain Constraints (Extended Model) s.t. min max d D y d y d {0,..., Λ } e E : alldifferent({y d p(d, e)}) Helmut Simonis Static Design RWA
53 Optimization from below Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Start with known lower bound Test value to see if problem is feasible If successful, optimal solution reached Otherwise increase bound by one and repeat Helmut Simonis Static Design RWA
54 Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Phase Finite Domain Simplified Extended Model y d {0,..., C} e E : alldifferent({y d p(d, e)}) Helmut Simonis Static Design RWA
55 SAT Model Problem Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model d D λ,λ Λ s.t. λ λ : x λ d x λ d d D : λ Λ x λ d e E λ Λ, d, d D s.t. p(d, e) p(d, e) d d : x λ d x λ d Helmut Simonis Static Design RWA
56 SAT Solver Problem Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Use minisat as black box Generate problem file in clausal format Impose external time limit (00 sec per run) Helmut Simonis Static Design RWA 6
57 Recall: Basic Clique Sizes Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Color Off Capacity 0 unused Helmut Simonis Static Design RWA 7
58 Problem Graph Coloring Solution Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Helmut Simonis Static Design RWA 8
59 Frequency Assignment Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Frequency Helmut Simonis Static Design RWA 9
60 Frequency Assignment Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Frequency Helmut Simonis Static Design RWA 60
61 Frequency Assignment Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Frequency Helmut Simonis Static Design RWA 6
62 Frequency Assignment Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Frequency Helmut Simonis Static Design RWA 6
63 Frequency Assignment Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Frequency Helmut Simonis Static Design RWA 6
64 Frequency Assignment Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Frequency Helmut Simonis Static Design RWA 6
65 Frequency Assignment Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Frequency Helmut Simonis Static Design RWA 6
66 Frequency Assignment Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Frequency Helmut Simonis Static Design RWA 66
67 Frequency Assignment Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Frequency Helmut Simonis Static Design RWA 67
68 Frequency Assignment Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Frequency Helmut Simonis Static Design RWA 68
69 Frequency Assignment Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Frequency Helmut Simonis Static Design RWA 69
70 Frequency Assignment Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Frequency Helmut Simonis Static Design RWA 70
71 Frequency Assignment Phase MIP Phase MIP Phase Finite Domain Model Phase SAT Model Frequency Helmut Simonis Static Design RWA 7
72 Outline Problem Basic Model Extended Problem Scalability Problem Basic Model Extended Problem Scalability Helmut Simonis Static Design RWA 7
73 Example Networks Problem Basic Model Extended Problem Scalability nsf nodes, edges eon 0 nodes, 78 edges mci 9 nodes, 6 edges brezil 7 nodes, 0 edges Helmut Simonis Static Design RWA 7
74 Basic Model Extended Problem Scalability Comparison (Basic Problem, 00 Runs Each) Complete Network Dem. MIP MIP-MIP MIP-FD MIP-SAT Opt Avg Opt Avg Opt Avg Opt Avg brezil brezil brezil brezil brezil brezil eon eon eon eon eon eon mci mci mci mci mci mci nsf nsf nsf nsf nsf nsf Helmut Simonis Static Design RWA 7
75 Basic Model Extended Problem Scalability Comparison (Extended Problem, 00 Runs Each) Complete Network Dem. MIP MIP-MIP MIP-FD MIP-SAT Opt Avg Opt Avg Opt Avg Opt Avg brezil brezil brezil brezil brezil brezil eon eon eon eon eon eon mci mci mci mci mci mci nsf nsf nsf nsf nsf nsf Helmut Simonis Static Design RWA 7
76 Basic Model Extended Problem Scalability Increasing Demand Number (Extended Problem, 00 Runs Each) Network Dem. λ Opt. Avg Avg Avg Max LP Max FD Avg MIP Max MIP Avg FD Max FD LP MIP FD Gap Gap Time Time Time Time brezil brezil brezil brezil brezil brezil brezil brezil brezil brezil brezil brezil brezil brezil Helmut Simonis Static Design RWA 76
77 Basic Model Extended Problem Scalability Increasing Network Size (Extended Problem, 00 Runs Each) Network Dem. λ Opt. Avg Avg Avg Max LP Max FD Avg MIP Max MIP Avg FD Max FD LP MIP FD Gap Gap Time Time Time Time r r r r Helmut Simonis Static Design RWA 77
78 Conclusions Outline Conclusions Helmut Simonis Static Design RWA 78
79 Conclusions Conclusions Modelling static design variant of RWA Choice of objective function important Simple MIP-FD decomposition works well Very good lower bound from phase MIP/LP FD graph coloring model outperforms MIP and SAT variants Possible to use specialized graph coloring codes (not tested) Conceptually simpler than RWA demand acceptance Helmut Simonis Static Design RWA 79
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