An Exact Algorithm for IP Traffic Engineering Andreas Bley Zuse - - PDF document

DFG Research Center MATHEON

Mathematics for key technologies

An Exact Algorithm for IP Traffic Engineering

Andreas Bley Zuse Institute Berlin

• Dept. Optimization

www.zib.de

• A. Bley, An Exact Algorithm for IP Traffic Engineering

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Overview

Cooperation with DFN-Verein

• German national research and education network
• connects over 700 universities, research labs, …
• ver 5.000 TByte traffic per month (2007)

I P Network Optimization

Long-term:

Network architecture

Mid-term:

Dimensioning

Short-term:

Traffic engineering Our Algorithm:

• ptimal (proven!!!) routing in backbone network
• incl. restoration paths, delay bounds, path restrictions, …

X-WiN IP topology, 02/2008

• A. Bley, An Exact Algorithm for IP Traffic Engineering

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Internet Routing

3 4 5 4 1 1 1 2 1 1 1 3 3 1 1 1 1

I ntra-Domain: Shortest Path Routing

(OSPF, IS-IS, RIP, …)

(1)

Assign routing weights to links

(2)

Send data along shortest paths

Routing control

• nly indirectly via lengths
• nly all paths together

Variants

• Distance-vector vs.

Link-state

• Single path

vs. Multi-path

Main challenges for planning

all paths depend on same routing metric no description of valid path sets without routing weights

• A. Bley, An Exact Algorithm for IP Traffic Engineering

4 Paketverlustrate auf Link

0.5 1 1.5 Auslastung Verlustrate

Link Congestion Delay

Traffic Engineering

TE planning problem (simplified) Given:

fixed network topology & capacities traffic demands (forecast/measurement)

Task: find/ change routing weights (only!!!) Goal: minimize maximum congestion Traffic Engineering:

Adjust routing to traffic demands

Goals: low packet loss rate low delay low jitter Key figure: link congestion

• A. Bley, An Exact Algorithm for IP Traffic Engineering

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Solution Methods I

... almost hopeless for real-world problem sizes. ... good solutions, but no or weak lower bounds.

Weight-based solution approaches

Modify weights Evaluate effects on routing

Local search, genetic algorithms, simulated annealing, …

[Bley+ 98, FortzThorup00, FortzThorup04, FortzÜmit07, Farago+ 98, Ericsson+ 02, Buriol+ 05, ...]

Lagrangian approaches

[LinWang93, Bley03, ...]

Flow-based approaches

Optimize end-to-end paths and compatible routing weights

Integrated MILP- or CP-Models

[Bourquia+ 03, DeGiovanniFortzLabbe05, PioroTomaszewski+ 05, ParmarAhmedSokol05, ...]

Obtained by linearizing quadratic models huge size, big-M coefficients, weak LP bounds

• A. Bley, An Exact Algorithm for IP Traffic Engineering

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Solution Methods II

... proven optimal solutions for real-world problem sizes. ... good solutions, but no or weak lower bounds.

Weight-based solution approaches

Modify weights Evaluate effects on routing

Local search, genetic algorithms, simulated annealing, …

[Bley+ 98, FortzThorup00, FortzThorup04, FortzÜmit07, Farago+ 98, Ericsson+ 02, Buriol+ 05, ...]

Lagrangian approaches

[LinWang93, Bley03, ...]

Flow-based approaches

Optimize end-to-end paths Find compatible weights

Decomposition

[B.00, B.Koch 02, HolmbergYuan01, Prytz02, B. 2007, PioroTomaszewski2007, ...]

Master: optimize end-to-end paths (integer programming) Client:

find compatible routing weights (linear programming)

• A. Bley, An Exact Algorithm for IP Traffic Engineering

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I ntegrated: Shortest path routing optimization Variables

routing weights path or arc-flows per demand link congestion

Constraints

link capacity constraints flow conservation and integrality shortest path routing

Decomposition Algorithm I

too many, too weak

• A. Bley, An Exact Algorithm for IP Traffic Engineering

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I nverse shortest paths problem Solved by linear programming

• Feasible: compatible weights

(scaling and rounding)

• Infeasible: violated constraint

(LP-dual and Greedy)

Single path routing problem

(with some extra constraints)

Solved by branch-and-cut

• Specialized branching rules
• Problem specific heuristics
• Additional strong cuts

Master: Routing path optimization Variables

routing weights path or arc-flows per demand link congestion

Constraints

link capacity constraints flow conservation and integrality shortest path routing (easy) shortest path routing (hard)

Client: Find compatible weights

(or viol. shortest path routing constraint)

Variables

routing weights

Constraints

paths are unique shortest paths

Decomposition Algorithm II

hard shortest path routing constraint Collection of routing paths

• A. Bley, An Exact Algorithm for IP Traffic Engineering

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Special Cuts

Clique inequalities for Bellman conflicts (subpath consistency) most important ‘easy’ shortest path constraints Rank inequalities for irreducible shortest path conflicts ‘hard’ shortest path constraints necessary and sufficient for correctness of model separation for (near-) integer routings via client problem I nduced knapsack cover inequalities exploit subpath consistency and integrality indispensible for good performance

Paths Precedences

P5 P4 P3 P2 P1 P6 P7

• A. Bley, An Exact Algorithm for IP Traffic Engineering

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Results I

DFN SNDlib

proven optimal solutions for small and medium-size networks very good solutions for large problems

real-world problems with symmetric single path routing and hop/delay restrictions,

• bjective values scaled, path-flow formulation, times in seconds on P4 3.2 GHz

• A. Bley, An Exact Algorithm for IP Traffic Engineering

11 benchmark instances from SNDlib with asymmetric demands, symmetric single path routing, no hop/delay restrictions; arc-flow formulation; times on P4 2.8 GHz

Results II

SNDlib

proven optimal solutions for small and medium-size networks very good solutions for large problems

• A. Bley, An Exact Algorithm for IP Traffic Engineering

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17,0 10 20 30 40 50 Congestion (in % ) 10 20 30 40 50 Congestion (in % ) 10 20 30 40 50 Congestion (in % )

Easy to implement new routing Immediate quality improvements More robust against traffic changes Less capacity expansion

Inverse capacities

Optimized weights: Lmax= 17%

10 20 30 40 50 Congestion (in % )

Geographic link lengths Unit weights

Default weights: Lmax> 38%

Practical Impact

Substantial load reduction by routing optimization!

Example: G-WiN 2 network

• A. Bley, An Exact Algorithm for IP Traffic Engineering

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Summary

Decomposition approach: MI LP for routing paths + LP for weights

Only practical method to compute proven optimal solutions Yields: optimal solutions for small & medium size problems best known solutions and bounds for large problems Applicable also for network topology and capacity planning

Software I mplementation at ZI B / atesio GmbH:

Variants and extensions:

Detailed node and link hardware model Hop limits, delay limits, path constraints Explicitly configured LSPs Routing in failure situations Several alternative MILP formulations

Successfully used in practice for many years

(German national research and education network)