1. March 2002Recents developments in metaheuristics and applications to network design problems Slide 1/59 (EHESS) Celso C. RIBEIRO Catholic University of Rio de Janeiro, Brazil Department of Computer Science Recent Developments in Metaheuristics and Applications to Network Design Problems Joint work with Maurício Resende and Isabel Rosseti

2. March 2002Recents developments in metaheuristics and applications to network design problems Slide 2/59 (EHESS) Summary 2-path network design problem PVC routing problem GRASP with path-relinking –Construction phase –Local search phase –Extended local search –Path-relinking Parallel implementation Numerical results –Variants of path-relinking –Effectiveness of the new heuristics –Parallelization Concluding remarks

3. March 2002Recents developments in metaheuristics and applications to network design problems Slide 3/59 (EHESS) 2-path network design problem Graph G = (V,E) V: node set E: edge set weights w e associated with each edge e  E k-path between nodes s,t  V: sequence of at most k edges connecting s and t D: set of demands (origin-destination pairs)

4. March 2002Recents developments in metaheuristics and applications to network design problems Slide 4/59 (EHESS) 2-path network design problem 2-path network design problem (2PNDP): Find a minimum weighted subset of edges E’  E containing a 2-path in G between the extremities of every origin-destination pair in D Applications: design of communication networks, in which paths with few edges are sought to enforce high reliability and small delays

5. March 2002Recents developments in metaheuristics and applications to network design problems Slide 5/59 (EHESS) 2-path network design problem Dahl & Johannessen (2000): –Decision version of 2PNDP is NP- complete. –Approximate algorithm –Exact cutting plane algorithm Balakrishnan & Altinkemer (1992): –Integer programming formulation for kPNDP –See also LeBlanc, Chifflet & Mahey (1999). Generalizations: k-hop minimum spanning tree, k-hop minimum Steiner tree

6. March 2002Recents developments in metaheuristics and applications to network design problems Slide 6/59 (EHESS) PVC routing Virtual private networks: permanent virtual circuits (PVCs) between customer endpoints on a backbone network Routing: either automatically by switch or by network designer without any knowledge of future requests Inefficiencies and occasional need for off-line rerouting of the PVCs

7. March 2002Recents developments in metaheuristics and applications to network design problems Slide 7/59 (EHESS) PVC routing: example

8. March 2002Recents developments in metaheuristics and applications to network design problems Slide 8/59 (EHESS) PVC routing: example

9. March 2002Recents developments in metaheuristics and applications to network design problems Slide 9/59 (EHESS) PVC routing: example

10. March 2002Recents developments in metaheuristics and applications to network design problems Slide 10/59 (EHESS) PVC routing: example

11. March 2002Recents developments in metaheuristics and applications to network design problems Slide 11/59 (EHESS) PVC routing: example max capacity = 3

12. March 2002Recents developments in metaheuristics and applications to network design problems Slide 12/59 (EHESS) PVC routing: example max capacity = 3 very long path! reroute

13. March 2002Recents developments in metaheuristics and applications to network design problems Slide 13/59 (EHESS) PVC routing: example max capacity = 3

14. March 2002Recents developments in metaheuristics and applications to network design problems Slide 14/59 (EHESS) PVC routing: example max capacity = 3 feasible and optimal!

15. March 2002Recents developments in metaheuristics and applications to network design problems Slide 15/59 (EHESS) PVC routing Other algorithms simply handle the number of hops (e.g. routing algorithm in Cisco switches) Handling delays is particularly important in international networks, where distances between backbone nodes vary considerably Cisco Catalystic 5505 switch

16. March 2002Recents developments in metaheuristics and applications to network design problems Slide 16/59 (EHESS) PVC routing Load balancing is important for providing flexibility to handle: –overbooking: typically used by network designers to account for non-coincidence of traffic –PVC rerouting: due to failures –bursting above the committed rate: not only allowed, but also sold to customers as one of the attractive features of frame relay Integer multicommodity network flow problem

17. March 2002Recents developments in metaheuristics and applications to network design problems Slide 17/59 (EHESS) PVC routing Given undirected FR network G = (V, E), where –V denotes n backbone nodes (FR switches) –E denotes m trunks connecting backbone nodes for each trunk e = (i,j ) –b (e ): maximum bandwidth (max kbits/sec rate) –c (e ): maximum number of PVCs that can be routed on it –d (e ): propagation and hopping delay

18. March 2002Recents developments in metaheuristics and applications to network design problems Slide 18/59 (EHESS) PVC routing Demands K = {1,…,p } defined by –Origin-destination pairs (o,d ) –r (p): effective bandwidth requirement (forward, backward, overbooking) for PVC p Objective is to minimize –delays –network load unbalance subject to –technological constraints

19. March 2002Recents developments in metaheuristics and applications to network design problems Slide 19/59 (EHESS) PVC routing route for PVC (o,d ) is a sequence of adjacent trunks from node o to node d set of routing assignments is feasible if for all trunks e –total bandwidth requirements routed on e does exceed b (e) –number of PVCs routed on e not greater than c(e)

20. March 2002Recents developments in metaheuristics and applications to network design problems Slide 20/59 (EHESS) Cost function Linear combination of –delay component - weighted by (1-  ) –load balancing component - weighted by  Delay component:

21. March 2002Recents developments in metaheuristics and applications to network design problems Slide 21/59 (EHESS) Cost function Load balancing component: measure of Fortz & Thorup (2000) to compute congestion:  =  1 (L 1 ) +  2 (L 2 ) + … +  |E| (L |E| ) where L e is the load on link e  E,  e (L e ) is piecewise linear and convex,  e (0) = 0, for all e  E.

22. March 2002Recents developments in metaheuristics and applications to network design problems Slide 22/59 (EHESS) Piecewise linear and convex  e (L e ) link congestion measure slope = 1 slope = 3slope = 10 slope = 70 slope = 500 slope = 5000 (Lece)(Lece)

23. March 2002Recents developments in metaheuristics and applications to network design problems Slide 23/59 (EHESS) GRASP with path-relinking GRASP: –Multistart metaheuristic, Feo & Resende (1989) Path-relinking: –Intensification strategy, Glover (1996) Repeat for MaxIterations: –Construct greedy randomized solution –Use local search to improve constructed solution –Apply path-relinking to further improve solution –Update pool of elite solutions –Update best solution found

24. March 2002Recents developments in metaheuristics and applications to network design problems Slide 24/59 (EHESS) Greedy Randomized Adaptive Search Procedures (GRASP) Combinaison d’une méthode constructive avec une approche par recherche locale, dans une procédure itérative (multi-départ) où les itérations sont totalement indépendantes les unes des autres (a) construction d’une solution (b) recherche locale

25. March 2002Recents developments in metaheuristics and applications to network design problems Slide 25/59 (EHESS) Greedy Randomized Adaptive Search Procedures (GRASP) f(s*)  +  for i = 1,…,MaxIterations do Construire une solution s en utilisant un algorithme glouton randomisé Appliquer une procédure de recherche locale à partir de s, pour obtenir la solution s’ if f(s’) < f(s*) then s*  s’ end-for

26. March 2002Recents developments in metaheuristics and applications to network design problems Slide 26/59 (EHESS) GRASP Phase de construction: construire une solution réalisable, un élément à la fois Chaque itération: –évaluer le bénéfice de chaque élément en utilisant une fonction gloutonne –créer une liste restricte de candidats, formée par les éléments avec les meilleures évaluations –sélectionner de façon probabiliste un élément de la liste restricte de candidats –adapter la fonction gloutonne après l’utilisation de l’élément choisi

27. March 2002Recents developments in metaheuristics and applications to network design problems Slide 27/59 (EHESS) GRASP Restriction des éléments dans la liste de candidats: –nombre maximum d’éléments dans la liste –qualité des éléments dans la liste (par rapport au choix exclusivement glouton) c min = bénefice minimum parmi tous les éléments c max = bénefice maximum parmi tous les éléments Paramètre α permet de controler la qualité des éléments dans la liste de candidats: Liste = { éléments j: c j ≤ c min + α. (c max - c min )} α = 0: choix glouton α = 1: choix probabiliste

28. March 2002Recents developments in metaheuristics and applications to network design problems Slide 28/59 (EHESS) GRASP Choix probabiliste entre les meilleurs éléments de la liste de candidats (pas forcément le meilleur, comme c’est le cas du choix exclusivement glouton) –la qualité moyenne de la solution dépend de la qualité des éléments dans la liste –la diversité des solutions construites dépend du nombre d’éléments dans la liste Diversification basée sur la randomisation controlée: différentes solutions construites lors de différentes iterations GRASP

29. March 2002Recents developments in metaheuristics and applications to network design problems Slide 29/59 (EHESS) GRASP Construction gloutonne: bonnes solutions (près des optima locaux), permettant d’accélérer la recherche locale Recherche locale: amélioration des solutions construites lors de la première phase –choix du voisinage –structures de données efficaces pour accélérer la recherche locale –bonnes solutions de départ permettent d’accélérer la recherche locale

30. March 2002Recents developments in metaheuristics and applications to network design problems Slide 30/59 (EHESS) GRASP Technique d’échantillonage de l’espace de recherche

31. March 2002Recents developments in metaheuristics and applications to network design problems Slide 31/59 (EHESS) GRASP Implémentation simple: algorithme glouton + recherche locale Peu de paramètres à régler: –restrictivité de la liste de candidats –nombre d’itérations Desavantage: n’utilise pas de mémoire Parallélisation: simple et directe –N itérations, p processeurs: chaque processeur fait N/p itérations et à la fin informe au maître la meilleure solution qu’il a trouvée

32. March 2002Recents developments in metaheuristics and applications to network design problems Slide 32/59 (EHESS) GRASP with extended local search Basic GRASP procedure puts too much effort in the construction phase, but stops in the first local optimum. Strategy to increase the effort in the local search phase and to add memory to GRASP. Improved local search by using a very short-term memory (tabu search).

33. March 2002Recents developments in metaheuristics and applications to network design problems Slide 33/59 (EHESS) GRASP with extended local search Keep a tabu list with the last TabuTenure (small value, typically 5- 10) solutions visited. Move to the best neighbor solution (be it an improving solution or not) which is not in the tabu list. Stop after TabuTenure iterations without improvement in the best global solution. Best results for capacitated minimum spanning tree: Souza, Duhamel & Ribeiro (2002)

34. March 2002Recents developments in metaheuristics and applications to network design problems Slide 34/59 (EHESS) GRASP for 2-path network design GRASP –Construction phase 1.Set the modified weights equal to the original weights. 2.Randomly select an origin-destination pair (a,b)  D. 3.Compute a shortest 2-path between a and b using the modified weights. 4.Set to 0 the modified weights of the edges in this path. 5.Remove (a,b) from D. 6.If D is empty stop, otherwise go back to step 2.

35. March 2002Recents developments in metaheuristics and applications to network design problems Slide 35/59 (EHESS) GRASP for 2-path network design GRASP –Local search phase 1.Generate a circular random permutation of the pairs in D. 2.Select the next origin-destination pair (a,b)  D. 3.Tentatively replace the shortest 2-path between a and b: Weights of edges used by other 2-paths are temporarilly set to 0. Compute a new shortest 2-path between a and b. Update the current solution if it is improved by the new 2-path. Restore all original edge weights. 4.If |D| paths have been investigated without improvement stop, otherwise go back to step 2.

36. March 2002Recents developments in metaheuristics and applications to network design problems Slide 36/59 (EHESS) Path-relinking Path-relinking: introduced in the context of tabu search by Glover (1996) –Intensification strategy using set of elite solutions Consists in exploring trajectories that connect high quality solutions. initial solution guiding solution path in neighborhood of solutions

37. March 2002Recents developments in metaheuristics and applications to network design problems Slide 37/59 (EHESS) Path-relinking Path is generated by selecting moves that introduce in the initial solution attributes of the guiding solution. At each step, all moves that incorporate attributes of the guiding solution are evaluated and the best move is taken: Initial solution guiding solution

38. March 2002Recents developments in metaheuristics and applications to network design problems Slide 38/59 (EHESS) Elite solutions x and y  (x,y): symmetric difference between x and y while ( |  (x,y)| > 0 ) { evaluate moves corresponding in  (x,y) make best move update  (x,y) } Path-relinking

39. March 2002Recents developments in metaheuristics and applications to network design problems Slide 39/59 (EHESS) GRASP with path-relinking Maintain an elite set of solutions found during GRASP iterations. After each GRASP iteration (construction and local search): –Select an elite solution at random: guiding solution. –Use GRASP solution as initial solution. –Perform path-relinking between these two solutions. PR allows introducing memory into GRASP.

40. March 2002Recents developments in metaheuristics and applications to network design problems Slide 40/59 (EHESS) GRASP with path-relinking P is a set of elite solutions. Each iteration of first |P| GRASP iterations adds one solution to P (if different from others). After that: solution x is promoted to P if: –x is better than best solution in P. –x is not better than best solution in P, but is better than worst and is sufficiently different from all solutions in P.

41. March 2002Recents developments in metaheuristics and applications to network design problems Slide 41/59 (EHESS) Path-relinking with GRASP Successful applications: –Prize-collecting Steiner tree problem Canuto, Resende & Ribeiro (2001) –Steiner tree problem Ribeiro, Uchoa & Werneck (2002) (e.g., best known results for open problems in series dv640 of the SteinLib) –Three-index assignment problem Aiex, Pardalos, Resende & Toraldo (2000) –Capacitated minimum spanning tree: Souza, Duhamel & Ribeiro (2002) (e.g., best known results for largest problems with 160 nodes)

42. March 2002Recents developments in metaheuristics and applications to network design problems Slide 42/59 (EHESS)

43. March 2002Recents developments in metaheuristics and applications to network design problems Slide 43/59 (EHESS) Parallel implementation Main interest of parallel implementations of metaheuristics: robustness Cung, Martins, Ribeiro & Roucairol (2001) Parallelization strategy: –Multiple-walk independent-thread strategy –Iterations evenly distributed over p processors –Each processor keeps a copy of the algorithm and data –One processor acts as the master (data, seeds, iterations) –Each processor performs Max_Iterations/p iterations

44. March 2002Recents developments in metaheuristics and applications to network design problems Slide 44/59 (EHESS) Computational results: 2-path network design Parallel GRASP heuristic: –Implementation in C –MPI LAM 6.3.2 for communication –Linux cluster with 32 Pentium II-400 processors Largest instances solved: –Larger instances solved with the GRASP heuristic: |V|= 400, |E|= 79800, |D|= 4000 (previously: |V|= 120, |E|= 7140, |D|= 60)

45. March 2002Recents developments in metaheuristics and applications to network design problems Slide 45/59 (EHESS) Computational results: 2-path network design Effectiveness: –100 small instances with 70 nodes generated as in Dahl and Johannessen (2000) for comparison purposes. –Statistical test t for unpaired observations –Parallel GRASP finds better solutions with 40% of confidence. Parall el GRAS P Sample A D&J (2000 ) Sample B Size10030 Mean443.7 (- 2.2%) 453.7 Std. dev. 40.661.6

46. March 2002Recents developments in metaheuristics and applications to network design problems Slide 46/59 (EHESS) Variants of GRASP with path- relinking: –GRASP: pure GRASP –G+PR(B): GRASP with backward PR –G+PR(F): GRASP with forward PR –G+PR(BF): GRASP with two-way PR Other strategies: –Truncated path-relinking –Do not apply PR at every iteration (frequency) Variants of GRASP with path- relinking S T T S S T S T

47. March 2002Recents developments in metaheuristics and applications to network design problems Slide 47/59 (EHESS) Variants of GRASP with path- relinking Select an instance and a target value. For each variant of GRASP with path- relinking: –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.

48. March 2002Recents developments in metaheuristics and applications to network design problems Slide 48/59 (EHESS) Variants of GRASP with path- relinking Each variant: 200 runs for one instance of 2PNDP

49. March 2002Recents developments in metaheuristics and applications to network design problems Slide 49/59 (EHESS) Variants of GRASP with path- relinking Same computation time: probability of finding a solution at least as good as the target value increases from GRASP  G+PR(F)  G+PR(B)  G+PR(BF) P(h,t) = probability that variant h finds a solution as good as the target value in time no greater than t –P(GRASP,10s) = 2% P(G+PR(F),10s) = 56% P(G+PR(B),10s) = 75% P(G+PR(BF),10s) = 84% Effectiveness of path-relinking to improve and speedup the pure GRASP

50. March 2002Recents developments in metaheuristics and applications to network design problems Slide 50/59 (EHESS) Computational results: PVC routing Heuristics: –H1: sorts demands in decreasing order and routes them using minimum hops paths –H2: sorts demands in decreasing order and routes using same cost function as GRASP –H3: adds the same local search to H2 –GPRb: GRASP with backwards path- relinking SGI Challenge 196 MHz

51. March 2002Recents developments in metaheuristics and applications to network design problems Slide 51/59 (EHESS) Computational results: PVC routing Test problems:

52. March 2002Recents developments in metaheuristics and applications to network design problems Slide 52/59 (EHESS) Variants of GRASP and path- relinking Time (s) Probability Each variant: 200 runs for one instance of PVC routing problem

53. March 2002Recents developments in metaheuristics and applications to network design problems Slide 53/59 (EHESS) Variants of GRASP and path- relinking Same computation time: probability of finding a solution at least as good as the target value increases from G  GPRf  GPRfb  GPRb P(h,t) = probability variant h finds solution as good as target value in time no greater than t –P(GPRfb,100s)=9.25% P(GPRb,100s)=28.75% –P(G,2000s)=8.33% P(GPRf,2000s)=65.25% P(h,time)=50% Times for each variant: –GPRb:129s G:10933s GPRf:1727s GPRfb:172s

54. March 2002Recents developments in metaheuristics and applications to network design problems Slide 54/59 (EHESS) Comparisons Distribution: 86/60/2: 86 edges with utilization in [0,1/3), 60 in [1/3,2/3), and two in [2/3,9/10) In general: GPRB > H3 > H2 > H1 (cost, max utilization, distribution) cost max util.

55. March 2002Recents developments in metaheuristics and applications to network design problems Slide 55/59 (EHESS) Parallel implementation: speedups Linear speedups: 2PNDP, |V|= 400, 3200 iterations

56. March 2002Recents developments in metaheuristics and applications to network design problems Slide 56/59 (EHESS) Concluding remarks (1/3) Effectiveness of the new heuristic for the 2-path network design problem: –Larger problems solved. –New heuristic finds better solutions. –Domination is stronger for harder or larger instances. Heuristic for PVC routing improves algorithm used in traffic engineering by network planners.

57. March 2002Recents developments in metaheuristics and applications to network design problems Slide 57/59 (EHESS) Concluding remarks (2/3) NETROUTER – Tool for optimally loading demands on single-path routes on a capacitated network. It uses the GPRb variant of the combination of GRASP and path-relinking, minimizing delays while balancing network load. Application - Netrouter is currently being used for the design of AT&T's next generation frame-relay and MPLS core architecture, to assess if the current and forecasted demands can be handled by the proposed trunking plan.

58. March 2002Recents developments in metaheuristics and applications to network design problems Slide 58/59 (EHESS) Concluding remarks (3/3) Path-relinking adds memory and intensification mechanisms to GRASP, systematically contributing to improve solution quality. Some implementation strategies appear to be more effective than others (e.g., backwards from better, elite solution to current locally optimal solution). Extended local search also adds memory and improves the solutions obtained by basic local search. Linear speedups with the parallel mplementation.

59. March 2002Recents developments in metaheuristics and applications to network design problems Slide 59/59 (EHESS) Slides and publications Slides of this talk can be downloaded from: http://www.inf.puc- rio/~celso/talks Papers about –recent developments and state-of-the-art on GRASP; –GRASP with path-relinking heuristic for PVC routing; –parallel GRASP heuristic for the 2-path network design problem; and –applications of GRASP and path-relinking on which this talk was based are available at: http://www.inf.puc- rio.br/~celso/publicacoes