August 2003 Tabu search heuristic to partition coloring1/36 MIC2003 MIC2003 Kyoto, August 25-28,...

August 2003 Tabu search heuristic to partition coloring1/36 MIC’2003

MIC’2003Kyoto, August 25-28, 2003 Bora Bora, Tahiti

August 2003 Tabu search heuristic to partition coloring2/36 MIC’2003

MIC’2003Kyoto, August 25-28, 2003

A Tabu Search Heuristic for Partition Coloring with an

Application to Routing and Wavelength Assignment

August 2003 Tabu search heuristic for partition coloring3/36 MIC’2003

MIC’2003Kyoto, August 25-28, 2003

A Tabu Search Heuristic for Partition Coloring with an

Application to Routing and Wavelength Assignment

Thiago NORONHA Celso C. RIBEIRO

Catholic University of Rio de JaneiroBrazil

August 2003 Tabu search heuristic to partition coloring4/36 MIC’2003

Introduction

• The partition coloring problem (PCP)• Routing and wavelength assignment in

all-optical networks (RWA)• Algorithms for PCP: construction, LS, tabu

search• Computational results• Application: static lightpath establishment• Conclusions

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Partition coloring problem (PCP)

• Graph G = (V,E) with vertex set partitioned into k disjoint subsets: V = V1 V2 ... Vp

• PCP consists in coloring exactly one node in each subset Vi , such that every two adjacent colored nodes have different colors.

• Objective: minimize the number of colors used.

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Partition coloring problem1

22

4

6

1

22

4

6

0

22

3

6

0

2

3

6

2

1 0

22

34

5

6

7

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Routing and wavelength assignment in circuit-switched

WDM all-optical networks• Different signals can be simultaneously

transmitted in a fiber, using different wavelengths: – Wavelength Division Multiplexing

• Connections (between origin-destination pairs) are established by lightpaths.

• To establish a lightpath consists in determining:– a route– a wavelength

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• Each signal can be switched optically at intermediate nodes in the network.

• No wavelength conversion is possible.• Lightpaths sharing a common link are not

allowed to use the same wavelength.• Traffic assumptions: Yoo & Banerjee

(1997)– static lightpath establishment– dynamic lightpath establishment

(O-D pairs are not known beforehand)

Routing and wavelength assignment in circuit-switched

WDM all-optical networks

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• Static lightpath establishment (SLE) without wavelength conversion:– Minimize the total number of used

wavelengths

– Other objective functions may also consider the load in the most loaded link, the total number of optical switches (total length), etc.

Routing and wavelength assignment in circuit-switched

WDM all-optical networks

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Optical network

Shortest path routing: three wavelengths are needed

Routing and wavelength assignment in circuit-switched

WDM all-optical networksFrom SLE to PCP

Lightpaths:A DB EC F

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Routing and wavelength assignment in circuit-switched

WDM all-optical networksFrom SLE to PCP Optical network

Lightpaths:A DB EC F

2-shortest path routing

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Routing and wavelength assignment in circuit-switched

WDM all-optical networksFrom SLE to PCP Optical network

Lightpaths:A DB EC F

2-shortest path routing: only two wavelengths are needed!

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Algorithms for PCP: Greedy heuristics

• Onestep Largest First• Onestep Smallest Last• Onestep Color Degree (onestepCD)

– best in literature: Li & Simha (2000)

• Twostep Largest First• Twostep Smallest Last• Twostep Color Degree

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1. Remove all edges whose vertices are in same group.

2. Find the vertex with minimal color-degree for each uncolored group.

3. Among these vertices, find that with the largest color-degree.

4. Assign to this vertex the smallest available color and remove all other vertices in the same group.

5. Repeat the above steps until all groups are colored.

Algorithms for PCP: OnestepCD

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1 0

3

2

4 5 6

7

8

CD: 0UD: 4CD: 0

UD: 3CD: 0UD: 2

CD: 0UD: 2

1 0

3

2

4 5 6

7

8

CD: 0UD: 3

CD: 0UD: 2

CD: 0UD: 2

0

3

2

4 5 6

7

8CD: 1UD: 0

CD: 1UD: 0

0

2

4 5 6

7

8

CD: 1UD: 0

0

2

4 5 6

8

0

2

6

8

Algorithms for PCP: OnestepCD

• Color degree: number of colored neighborsUncolored degree: number of uncolored neighbors

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• First, LS-PCP converts a feasible solution with C colors into an infeasible solution with C-1 colors; next, it attempts to restore solution feasibility.

• The local search procedure investigates the subsets whose colored node is involved in a coloring conflict.

• LS-PCP searches within each subset for a node that can be colored or recolored so as to reduce the overall number of coloring conflicts.

Algorithms for PCP: Local search (1/2)

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• In case such a node exists, the algorithm moves to a new solution. Otherwise, another subset is randomly chosen and investigated.

• If a feasible solution with C-1 colors is found, the feasibility of this coloring is destroyed and another coloring using C-2 colors is sought.

• LS-PCP stops when the number of coloring conflicts cannot be reduced and the solution is still infeasible.

Algorithms for PCP: Local search (2/2)

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1 0

3

2

4 5 6

7

8

1 0

3

2

4 5 6

7

8

1 0

3

2

4 5 6

7

8

1 0

3

2

4 5 6

7

8

Algorithms for PCP: Local search

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• Simple short-term memory strategy: TS-PCP

• Initial solutions: OnestepCD• Local search strategy: LS-PCP

– move: pair (node,color)

• Tabu tenure: randomly in U[C/4,3C/4]• Aspiration criterion: improve best• Stopping criterion: C.P.10 iterations

without finding a feasible solution, where C = number of colors and P = number of subsets in the partition

Algorithms for PCP: Tabu search

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Computational results

• Random instances: – eight PCP instances generated from graph

coloring instances DJSC-250.5 and DJSC-500.5Aragon, Johnson, McGeoch & C. Schevon (1991)• nodes in original instance are replicated (2x, 3x, 4x)• edges are additioned with density 0.5• one subset for each original node

• Computational experiments: Pentium IV 2.0 GHz

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Computational resultsAverage results: construction, local search, tabu search

OnestepCD

Local search

Tabu search

Instance nodes

colors colors

% red.

colors

% red.

DSJC-250.5-1

250 41.7 40.6 3 29.6 29

DSJC-250.5-2

500 40.4 38.1 6 25.8 36

DSJC-250.5-3

750 38.8 35.6 8 24.0 38

DSJC-250.5-4

1000 38.3 34.7 9 23.0 40

DSJC-500.5-1

500 71.2 69.3 3 52.6 26

DSJC-500.5-2

1000 69.5 67.3 3 46.6 33

DSJC-500.5-3

1500 68.8 65.4 5 43.9 36

DSJC-500.5-4

2000 68.7 62.5 9 42.4 38

6% 35%

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Computational resultsTabu search: solution values and times (10 runs)Colors Time (s)

Instance bestavera

geworst

to best

total

DSJC-250.5-1

29 29.6 30 6.7 21.4

DSJC-250.5-2

25 25.8 26 11.7 62.4

DSJC-250.5-3

24 24.0 24 35.2 164.7

DSJC-250.5-4

23 23.0 23 65.3 300.8

DSJC-500.5-1

52 52.6 53 41.9 197.2

DSJC-500.5-2

46 46.6 47286.

51068.

3

DSJC-500.5-3

43 43.9 44533.

82187.

5

DSJC-500.5-4

42 42.4 43777.

73349.

6

Robust!

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Computational resultsRandom instances: varying the

number of subsets

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Computational resultsRandom instances: varying the graph

density

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• Select an instance and a target value:– Perform 200 runs using different seeds.– Stop when a solution value at least as good as

the target is found.– For each run, measure the time-to-target-value.– Plot the probabilities of finding a solution at

least as good as the target value within some computation time.

• Plots can illustrate algorithm robustness and are very useful for comparisons based on the probability distribution of the time-to-target-value– Aiex, Resende & Ribeiro (2002) – Resende & Ribeiro (2003, this meeting)

Time-to-target-value plots

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Instance DSJC-250.5-4

Time-to-target-value plots

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• Possible routing algorithms:– k-shortest paths– Path stripping: solves LP relaxation and

builds progressively longer shortest routes using edges in the fractional solution.Banerjee & Mukherjee (1995)

– Greedy-EDP-RWA: multistart construction using random permutations (greedy max edge-disjoint paths routing), too many restarts are needed.Manohar, Manjunath & Shevgaonkar (2002)

Static Lightpath Establishment

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• Comparison:– n-Greedy-EDP-RWA vs. ...– ... two routing iterations of Greedy-EDP-

RWA followed by partition coloring using TS-PCP

• Both algorithms stop when a target solution value is found:– Target is the optimal value of the LP

relaxation of the IP formulation without optical continuity constraints.

Application: SLE

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SLE instance #1: 14 nodes, 21 links, and 182

connections

Application: SLE

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SLE instance #1: target = 13 (optimal)

Application: SLE

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Application: SLESLE instance #2:

27 nodes, 70 links, and 702 connections

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Application: SLESLE instance #2: target = 24

(optimal)

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Conclusions

• We proposed a local search procedure and a tabu search heuristic for partition coloring.

• TS-PCP is able to significantly improve the solutions obtained by OnestepCD.

• TS-PCP together with a routing algorithm can be successfully used to solve SLE in RWA.

• Future work will consider other routing algorithms to be used with TS-PCP to solve SLE in practical applications.

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Slides and publications

• Slides of this talk can be downloaded from: http://www.inf.puc-rio/~celso/talks

• Paper will be soon available at:http://www.inf.puc-rio.br/~celso/publicacoes

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Announcements

• IV Workshop on Efficient and Experimental AlgorithmsBúzios (Brazil), May 25 to 28, 2004

IV Workshop on Efficient and Experimental AlgorithmsBúzios (Brazil), May 25 to 28, 2004

August 2003 Tabu search heuristic to partition coloring36/36 MIC’2003

XIX International Symposium on Mathematical Programming

Rio de Janeiro (Brazil), July 2006