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

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August 2003 Tabu search heuristic to partition coloring2/36 MIC2003 MIC2003 Kyoto, August 25-28, 2003 A Tabu Search Heuristic for Partition Coloring with an Application to Routing and Wavelength Assignment

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August 2003 Tabu search heuristic for partition coloring3/36 MIC2003 MIC2003 Kyoto, 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 Janeiro Brazil

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August 2003 Tabu search heuristic to partition coloring4/36 MIC2003 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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August 2003 Tabu search heuristic to partition coloring5/36 MIC2003 Partition coloring problem (PCP) Graph G = (V,E) with vertex set partitioned into k disjoint subsets: V = V 1 V 2... V p PCP consists in coloring exactly one node in each subset V i, such that every two adjacent colored nodes have different colors. Objective: minimize the number of colors used.

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August 2003 Tabu search heuristic to partition coloring6/36 MIC2003 Partition coloring problem 1 22 4 6 1 22 4 6 0 22 3 6 0 2 3 6 2 10 22 3 4 5 6 7

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August 2003 Tabu search heuristic to partition coloring7/36 MIC2003 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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August 2003 Tabu search heuristic to partition coloring8/36 MIC2003 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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August 2003 Tabu search heuristic to partition coloring9/36 MIC2003 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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August 2003 Tabu search heuristic to partition coloring10/36 MIC2003 Optical network Shortest path routing: three wavelengths are needed Routing and wavelength assignment in circuit-switched WDM all-optical networks From SLE to PCP Lightpaths: A D B E C F

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August 2003 Tabu search heuristic to partition coloring11/36 MIC2003 Routing and wavelength assignment in circuit-switched WDM all-optical networks From SLE to PCP Optical network Lightpaths: A D B E C F 2-shortest path routing

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August 2003 Tabu search heuristic to partition coloring12/36 MIC2003 Routing and wavelength assignment in circuit-switched WDM all-optical networks From SLE to PCP Optical network Lightpaths: A D B E C F 2-shortest path routing: only two wavelengths are needed!

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August 2003 Tabu search heuristic to partition coloring13/36 MIC2003 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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August 2003 Tabu search heuristic to partition coloring14/36 MIC2003 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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August 2003 Tabu search heuristic to partition coloring15/36 MIC2003 CD: 0 UD: 4 CD: 0 UD: 3 CD: 0 UD: 2 CD: 0 UD: 2 CD: 0 UD: 3 CD: 0 UD: 2 CD: 0 UD: 2 CD: 1 UD: 0 CD: 1 UD: 0 CD: 1 UD: 0 Algorithms for PCP: OnestepCD Color degree: number of colored neighbors Uncolored degree: number of uncolored neighbors

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August 2003 Tabu search heuristic to partition coloring16/36 MIC2003 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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August 2003 Tabu search heuristic to partition coloring17/36 MIC2003 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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August 2003 Tabu search heuristic to partition coloring18/36 MIC2003 Algorithms for PCP: Local search

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August 2003 Tabu search heuristic to partition coloring19/36 MIC2003 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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August 2003 Tabu search heuristic to partition coloring20/36 MIC2003 Computational results Random instances: –eight PCP instances generated from graph coloring instances DJSC-250.5 and DJSC-500.5 Aragon, 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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August 2003 Tabu search heuristic to partition coloring21/36 MIC2003 Computational results Average results: construction, local search, tabu search Onestep CD Local search Tabu search Instancenode s colors % red. color s % red. DSJC- 250.5-1 25041.740.6329.629 DSJC- 250.5-2 50040.438.1625.836 DSJC- 250.5-3 75038.835.6824.038 DSJC- 250.5-4 100038.334.7923.040 DSJC- 500.5-1 50071.269.3352.626 DSJC- 500.5-2 100069.567.3346.633 DSJC- 500.5-3 150068.865.4543.936 DSJC- 500.5-4 200068.762.5942.438 6% 35%

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August 2003 Tabu search heuristic to partition coloring22/36 MIC2003 Computational results Tabu search: solution values and times (10 runs) ColorsTime (s) Instancebest averag e worst to best total DSJC- 250.5-1 2929.6306.721.4 DSJC- 250.5-2 2525.82611.762.4 DSJC- 250.5-3 2424.02435.2164.7 DSJC- 250.5-4 2323.02365.3300.8 DSJC- 500.5-1 5252.65341.9197.2 DSJC- 500.5-2 4646.647286.51068.3 DSJC- 500.5-3 4343.944533.82187.5 DSJC- 500.5-4 4242.443777.7 3349. 6 Robust!

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August 2003 Tabu search heuristic to partition coloring23/36 MIC2003 Computational results Random instances: varying the number of subsets

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August 2003 Tabu search heuristic to partition coloring24/36 MIC2003 Computational results Random instances: varying the graph density

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August 2003 Tabu search heuristic to partition coloring25/36 MIC2003 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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August 2003 Tabu search heuristic to partition coloring26/36 MIC2003 Instance DSJC-250.5-4 Time-to-target-value plots

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August 2003 Tabu search heuristic to partition coloring27/36 MIC2003 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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August 2003 Tabu search heuristic to partition coloring28/36 MIC2003 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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August 2003 Tabu search heuristic to partition coloring29/36 MIC2003 SLE instance #1: 14 nodes, 21 links, and 182 connections Application: SLE

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August 2003 Tabu search heuristic to partition coloring30/36 MIC2003 SLE instance #1: target = 13 (optimal) Application: SLE

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August 2003 Tabu search heuristic to partition coloring31/36 MIC2003 Application: SLE SLE instance #2: 27 nodes, 70 links, and 702 connections

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August 2003 Tabu search heuristic to partition coloring32/36 MIC2003 Application: SLE SLE instance #2: target = 24 (optimal)

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August 2003 Tabu search heuristic to partition coloring33/36 MIC2003 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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August 2003 Tabu search heuristic to partition coloring34/36 MIC2003 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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August 2003 Tabu search heuristic to partition coloring35/36 MIC2003 Announcements IV Workshop on Efficient and Experimental Algorithms Búzios (Brazil), May 25 to 28, 2004 IV Workshop on Efficient and Experimental Algorithms Búzios (Brazil), May 25 to 28, 2004

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August 2003 Tabu search heuristic to partition coloring36/36 MIC2003 XIX International Symposium on Mathematical Programming Rio de Janeiro (Brazil), July 2006

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