Novembro 2003 Tabu search heuristic for partition coloring1/29 XXXV SBPO

XXXV SBPONatal, 4-7 de novembro de 2003

A Tabu Search Heuristic for Partition Coloring with an

Application to Routing and Wavelength Assignment

Thiago NORONHA Celso C. RIBEIRO

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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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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 (greedy)

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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

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4 5 6

7

8

1 0

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4 5 6

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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

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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• 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)

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

• Local search and 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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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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Computational resultsRandom instances: varying the

number of subsets

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Computational resultsRandom instances: varying the graph

density