Martin Grötschel Institut für Mathematik, Technische Universität Berlin (TUB)DFG-Forschungszentrum “Mathematik für Schlüsseltechnologien” (MATHEON)Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)
[email protected] http://www.zib.de/groetschel
05M2 LectureTelecommunication
Network Design
Martin Grötschel Beijing Block Course
“Combinatorial Optimization at Work”September 25 – October 6, 2006
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CO atWork Contents
1. Telecommunication: The General Problem2. Newspaper Reports3. Survivability4. Integrated Topology, Capacity, and Routing
Optimization as well as Survivability Planning
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CO atWork Contents
1. Telecommunication: The General Problem2. Newspaper Reports3. Survivability4. Integrated Topology, Capacity, and Routing
Optimization as well as Survivability Planning
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CO atWork What is the Telecom Problem?
Design excellent technical devicesand a robust network that survivesall kinds of failures and organizethe traffic such that high qualitytelecommunication betweenvery many individual units at many locations is feasibleat low cost!
SpeechData
VideoEtc.
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CO atWork What is the Telecom Problem?
Design excellent technical devicesand a robust network that survivesall kinds of failures and organizethe traffic such that high qualitytelecommunication betweenvery many individual units at many locations is feasibleat low cost!
ThisThis problemproblem is is tootoo generalgeneral
to to bebe solvedsolved in in oneone stepstep..
Approach in Practice:• Decompose whenever possible.• Look at a hierarchy of problems.• Address the individual problems one by one.• Recompose to find a good global solution.
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CO atWork Connecting Mobiles: What´s up?
BSC
MSC
BSC
BSC
BSC
BSC
BSC
BSC
MSC
MSCMSC
MSC
BTS
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CO atWork Contents
1. Telecommunication: The General Problem2. Newspaper Reports3. Survivability4. Integrated Topology, Capacity, and Routing
Optimization as well as Survivability Planning
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CO atWork Clyde Monma
Cornell & Cayuga Lake 1987
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CO atWork USA 1987-1988
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CO atWork USA 1987-1988
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CO atWork Special Report by IEEE Spectrum
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CO atWork
Special Report
IEEE Spectrum
June1988
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CO atWork Berlin 1994 & Köln 1994
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CO atWork
High-TechTerrorism 1995
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CO atWork Berlin 1997 & Wien
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CO atWork Austria
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CO atWork Contents
1. Telecommunication: The General Problem2. Newspaper Reports3. Survivability4. Integrated Topology, Capacity, and Routing
Optimization as well as Survivability Planning
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CO atWork
Network Design: Tasks to be solvedSome Examples
Locating the sites for antennas (TRXs) and base transceiver stations (BTSs)
Assignment of frequencies to antennas
Cryptography and error correcting encoding for wirelesscommunication
Clustering BTSs
Locating base station controllers (BSCs)
Connecting BTSs to BSCs
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CO atWork
Network Design: Tasks to be solvedSome Examples (continued)
Locating Mobile Switching Centers (MSCs)
Clustering BSCs and Connecting BSCs to MSCs
Designing the BSC network (BSS) and theMSC network (NSS or core network)
Topology of the network
Capacity of the links and components
Routing of the demand
Survivability in failure situations
Most of these problems turn out to beCombinatorial Optimization or
Mixed Integer Programming Problems
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CO atWork
TheBellCorestudy
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CO atWork
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CO atWork The Data
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CO atWork The IP Model
minimum cost spanning treeminimum cost Steiner treemin-cost k-edge or k-node-connected subgraph
Special cases:
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CO atWork
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CO atWork
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CO atWork
Nodetypes
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CO atWork
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CO atWork b-matchings and r-covers
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CO atWork Generalizations of r-cover inequalities
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CO atWork Facets: one example
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CO atWork
Facets: anotherexample
≥
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CO atWork Data
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CO atWork Reduction
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CO atWork computational Results
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CO atWork LATA DL: optimum solution
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CO atWork The Ship Problem: higher connectivity
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CO atWork The Ship Problem: higher connectivity
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CO atWork The Ship Problem: higher connectivity
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CO atWork Contents
1. Telecommunication: The General Problem2. Newspaper Reports3. Survivability4. Integrated Topology, Capacity, and Routing
Optimization as well as Survivability Planning
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CO atWork
Network Design: Tasks to be solvedSome Examples (continued)
Locating Mobile Switching Centers (MSCs)
Clustering BSCs and Connecting BSCs to MSCs
Designing the BSC network (BSS) and theMSC network (NSS or core network)
Topology of the network
Capacity of the links and components
Routing of the demand
Survivability in failure situations
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CO atWork Network Optimization
Networks
Capacities Requirements
Cost
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CO atWork What needs to be planned?
TopologyCapacitiesRoutingFailure Handling (Survivability)
IP RoutingNode Equipment PlanningOptimizing Optical Links and Switches
DISCNET: A Network Planning Tool(Dimensioning Survivable Capacitated NETworks)
atesio ZIB Spin-Off
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CO atWork The Network Design Problem
200
65
258
134
30
42Düsseldorf
Frankfurt
Berlin
Hamburg
München
Communication Demands
Düsseldorf
Frankfurt
Berlin
Hamburg
MünchenPotential topology
&Capacities
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CO atWork Capacities
PDH
2 Mbit/s
34 Mbit/s
140 Mbit/s
SDH
155 Mbit/s
622 Mbit/s
2,4 Gbit/s
... WDM (n x STM-N)
Two capacity models : Discrete Finite CapacitiesDivisible Capacities
(P)SDH=(poly)synchronous digital hierarchy
WDM=Wavelength Division MultiplexerSTM-N=Synchronous Transport Modul with N STM-1 Frames
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CO atWork Survivability
H B
D
FM
120H B
D
FM
3030
60Diversification„route node-disjoint“
Path restoration„reroute affected demands“(or p% of all affected demands)
6060
H B
D
FM
H B
D
FM
H B
D
FM
60
60
Reservation„reroute all demands“(or p% of all demands)
120
H B
D
FM
H B
D
FM
H B
D
FM
60
60
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CO atWork Model: Capacities
Capacity variables : e ∈ E, t = 1, ..., Te
{0,1}tex ∈
Cost function :
1
mineT
t te e
e E t
k x∈ =∑ ∑
Capacity constraints : e ∈ E0 11 0eTe e ex x x= ≥ ≥ ≥ ≥
0
eTt t
e e et
y c x=
= ∑
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CO atWork Model: Routings
Path variables :
Capacity constraints :
Demand constraints :
, , ss uvs S uv D P∈ ∈ ∈ Ρ
( ) 0suvf P ≥
0
0
:
( )uv
e uvuv D P e P
y f P∈ ∈Ρ ∈
≥ ∑ ∑e E∈
0
0 ( )uv
uv uvP
d f P∈Ρ
= ∑uv D∈
Path length restriction
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CO atWork Model: Survivability (one example)
Path restoration „reroute affected demands“
H B
D
F
M
120
60
60
H B
D
F
M
0
0
s suv uv uv
suv uv uv uv
P Ρ Ρ P Ρ :e P
f (P) f (P) dσ∈ ∩ ∈ ∈
+ ≥∑ ∑for all s∈S, uv∈Ds
0
0(s ss uv uv uv
suv uv e
uv D P Ρ Ρ :e P P Ρ :e P
f (P) f (P) ) y∈ ∈ ∩ ∈ ∈ ∈
+ ≤∑ ∑ ∑for all s∈S, e∈Es
60
60
H B
D
F
M
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CO atWork Mathematical Model
topology decisison
capacity decisions
normal operation routing
component failure routing
suvs PDuvSs Ρ∈∈∈ ,,0)( ≥Pf suv
)(:
0
0
PfyDuv PeP
uveuv
∑ ∑∈ ∈Ρ∈
≥ Ee∈
∑Ρ∈
=0
)(0
uvPuvuv Pfd Duv ∈
∈ =∑ ∑
1
m ineT
t te e
e E t
k x
, 1, , ee E t T∈ = …
∈ = …, 1, , ee E t T∈ {0,1}tex
1t te ex x− ≥
0
eTt t
e e et
y c x=
= ∑ e E∈0
0
s suv uv uv
suv uv uv uv
P Ρ Ρ P Ρ :e P
f (P) f (P) dσ∈ ∩ ∈ ∈
+ ≥∑ ∑
0
0
s ss uv uv uv
suv uv e
uv D P Ρ Ρ :e P P Ρ :e P
( f (P) f (P)) y∈ ∈ ∩ ∈ ∈ ∈
+ ≤∑ ∑ ∑ sEeSs ∈∈ ,
sDuvSs ∈∈ ,
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CO atWork Flow chart
No
Yes
Separationalgorithms
Optimalsolution
InitializeLP-relaxation
Runheuristics
SolveLP-relaxation
Separationalgorithms
AugmentLP-relaxation
Solve feasibilityproblem
Inequalities?
x variablesinteger?
Yes
No
Feasibleroutings?
Yes
No
Polyhedral combinatoricsValid inequalities (facets)Separation algorithmsHeuristicsFeasibility of a capacity vector
LP-based approach:
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CO atWork Finding a Feasible Solution?
HeuristicsLocal search
Simulated Annealing
Genetic algorithms
...
Nodes Edges Demands Routing-Paths
15 46 78 > 150 x 10e6
36 107 79 > 500 x 10e9
36 123 123 > 2 x 10e12
Manipulation ofRoutingsTopologyCapacities
Problem Sizes
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CO atWork How much to save?
Real scenario• 163 nodes• 227 edges• 561 demands
34% potential savings!==
> hundred million dollars
PhD Thesis: http://www.zib.de/wessaely
Martin Grötschel Institut für Mathematik, Technische Universität Berlin (TUB)DFG-Forschungszentrum “Mathematik für Schlüsseltechnologien” (MATHEON)Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)
[email protected] http://www.zib.de/groetschel
05M2 LectureTelecommunication
Network Design
Martin Grötschel Beijing Block Course
“Combinatorial Optimization at Work”September 25 – October 6, 2006The End
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