Intra-domain weight optimization using column generation Bernard - - PowerPoint PPT Presentation

10th Aussois Workshop on Combinatorial Optimization, Aussois 2006

Intra-domain weight

• ptimization using

column generation

Bernard Fortz and Hakan Umit Université Catholique de Louvain Institut d'Administration et de Gestion Louvain-la-Neuve Belgium

2

Challenges

 Rapid growth of networks  Meeting user demands  Quality of service under service

level agreements; less delay, promised throughput etc.

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Intra-domain routing protocols

 Interior Gateway Protocol  OSPF, IS-IS  Routing information is distributed between routers

belonging to a single Autonomous System.

 Traffic is routed through shortest paths wrt link

weights

 Weights are set and can be altered by network

• perators

 Suggestion of Cisco: weight=1/capacity

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Packet routing in OSPF

2 1 3 4

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IGP weight optimization problem

 Find the best set of link metric (weights)

that yields routing of a given traffic (demands between routers) with minimum congestion (load over the links). Constraint:

 A flow arriving at a router (node) is sent

to its destination by evenly splitting the flow between the links that are on the shortest paths to the destination.

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Evenly balancing flows

NP-Hard!

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Existing approaches and tools

 Weight optimization using local search heuristic

[Fortz and Thorup, 2004]

 Tabu search implementation  IGP-WO: open source software, [Fortz, Cerav and

Umit, 2004]

 Open source software funded by Walloon

government

 Three research groups from UCL and Univ. Liege

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About the toolbox

Unified algorithms for intra-domain and inter- domain traffic engineering purposes Project URL : http://totem.info.ucl.ac.be

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Results so far

Results are within 5% gap of General routing problem

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Objectives

 Provide a lower bound  Generate possible link weights

for IGP routing by using column generation

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Problem input and variables

 with capacitated arcs,  commodities  demand for each and  Set of directed paths

Objective: Minimize total cost of flows: Decision variables:

 : Flow on path  : Load on arc

a

c

K k

k

F

N

• k

N dk

k

P

p

f

k

P p

a

l

A a

) , ( A N G =

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Piecewise linear cost function

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Multi commodity network flow problem – path based form

• A

a a

Minimize

:

• p

k P p p p a P p p a a i a i a

f F f f l c l

k k

• ,

, , , , ,

k

P p K k K k A a I i A a

• )

4 ( ) 3 ( ) 2 ( ) 1 (

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Column generation procedure

Constraints Variables that were never considered Initial variables Added variables Restricted Master Problem (RMP) Multi commodity flow problem- path based

paths

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Solving master and restricted master problem

Let be a subset of

 Solve restricted master problem for

paths in

 Add more columns as needed until

• ptimum solution is attained

k

S

k

P

k

S

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Optimality Conditions

 Dual variables

 for each arc  for each commodity

a

w

k

• :
• p

k P p p p a P p p a a i a i a

f F f f l c l

k k

• ,

, , , , ,

k

P p K k K k A a I i A a

• )

4 ( ) 3 ( ) 2 ( ) 1 (

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Interpretation of dual variables

 Dual Variables and are the optimal

weight for arc a and shortest path distance for commodity k, respectively.

 In column generation procedure dual

variables are input to check optimality:

a

w

k

• a

w

• <

p a k a P p

w

k

• min

Newly generated path Current path

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Generation of new columns

 Dynamic shortest path computation

[Buriol, Resende and Thorup, 2003]

 Given a graph , a shortest

path graph and a vector W with a weight associated with each arc a. Update without recomputing it from scratch.

 Gain up to factor of 20 for a 100 node

graph.

) , ( A N G = ) ' , ( A N GSP =

a

w

SP

G

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Use of output

 Solving RMP until optimality, i.e.

until no more shortest path exists, can provide a lower bound for IGP routing

 of the optimum solution can be

used as a heuristic weight setting

a

w

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Future research

 Numerical results  Addition of a cut that will split the

flows evenly