XXXV SBPO Natal , 4-7 de novembro de 200 3

1. XXXV SBPO Natal, 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 Tabu search heuristic for partition coloring

2. 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 Tabu search heuristic for partition coloring

3. 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. Tabu search heuristic for partition coloring

4. 1 1 2 2 2 2 6 6 1 0 4 4 7 2 2 0 0 6 3 5 2 2 2 2 4 6 6 3 3 Partition coloring problem Tabu search heuristic for partition coloring

5. Routing and wavelength assignment in circuit-switched WDM all-optical networks • Different signalscan 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 Tabu search heuristic for partition coloring

6. Routing and wavelength assignment in circuit-switched WDM all-optical networks • 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) Tabu search heuristic for partition coloring

7. Routing and wavelength assignment in circuit-switched WDM all-optical networks • 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. Tabu search heuristic for partition coloring

8. Shortest path routing: three wavelengths are needed Routing and wavelength assignment in circuit-switched WDM all-optical networks Optical network From SLE to PCP Lightpaths: A  D B  E CF Tabu search heuristic for partition coloring

9. Routing and wavelength assignment in circuit-switched WDM all-optical networks Optical network From SLE to PCP Lightpaths: A  D B  E CF 2-shortest path routing Tabu search heuristic for partition coloring

10. 2-shortest path routing: only two wavelengths are needed! Routing and wavelength assignment in circuit-switched WDM all-optical networks Optical network From SLE to PCP Lightpaths: A  D B  E CF Tabu search heuristic for partition coloring

11. Algorithms for PCP: OnestepCD (greedy) • Remove all edges whose vertices are in same group. • Find the vertex with minimal color-degree for each uncolored group. • Among these vertices, find that with the largest color-degree. • Assign to this vertex the smallest available color and remove all other vertices in the same group. • Repeat the above steps until all groups are colored. Tabu search heuristic for partition coloring

12. CD: 0 UD: 4 CD: 0 UD: 3 CD: 0 UD: 3 CD: 0 UD: 2 CD: 0 UD: 2 CD: 1 UD: 0 CD: 0 UD: 2 CD: 0 UD: 2 CD: 1 UD: 0 CD: 1 UD: 0 Algorithms for PCP: OnestepCD • Color degree: number of colored neighborsUncolored degree: number of uncolored neighbors Tabu search heuristic for partition coloring

13. Algorithms for PCP: Local search (1/2) • First, LS-PCP converts a feasible solution with C colors into an infeasible solution with C-1 colors; next, it attempts to restoresolution 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 recoloredso as toreduce the overall number of coloring conflicts. Tabu search heuristic for partition coloring

14. Algorithms for PCP: Local search (2/2) • In case such a node exists, the algorithm moves to a new solution.Otherwise, anothersubset 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. Tabu search heuristic for partition coloring

15. Algorithms for PCP: Local search Tabu search heuristic for partition coloring

16. Algorithms for PCP: Tabu search • 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 Tabu search heuristic for partition coloring

17. 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 Tabu search heuristic for partition coloring

18. 35% Computational results Average results: construction, local search, tabu search Tabu search heuristic for partition coloring

19. Computational results Tabu search: solution values and times (10 runs) Robust! Tabu search heuristic for partition coloring

20. Time-to-target-value plots • 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 robustnessand are very useful for comparisons based on the probability distribution of the time-to-target-value • Aiex, Resende & Ribeiro (2002) • Resende & Ribeiro (2003) Tabu search heuristic for partition coloring

21. Time-to-target-value plots Instance DSJC-250.5-4 Tabu search heuristic for partition coloring

22. Static Lightpath Establishment • 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) Tabu search heuristic for partition coloring

23. Application: SLE • 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. Tabu search heuristic for partition coloring

24. Application: SLE SLE instance #1: 14 nodes, 21 links, and 182 connections Tabu search heuristic for partition coloring

25. Application: SLE SLE instance #1: target = 13 (optimal) Tabu search heuristic for partition coloring

26. Application: SLE SLE instance #2: 27 nodes, 70 links, and 702 connections Tabu search heuristic for partition coloring

27. Application: SLE SLE instance #2: target = 24 (optimal) Tabu search heuristic for partition coloring

28. 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. Tabu search heuristic for partition coloring

29. 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 Tabu search heuristic for partition coloring

30. 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 Tabu search heuristic for partition coloring

31. Computational results Random instances: varying the number of subsets Tabu search heuristic for partition coloring

32. Computational results Random instances: varying the graph density Tabu search heuristic for partition coloring