1. February 2002GRASP with path-relinking for PVC routingSlide 1/42 (ROADEF) A GRASP with path- relinking heuristic for PVC routing Celso C. Ribeiro Computer Science Department Catholic University of Rio de Janeiro joint work with M.G.C. Resende

2. February 2002GRASP with path-relinking for PVC routingSlide 2/42 (ROADEF) Summary PVC routing Integer multicommodity flow formulation Cost function Solution method: GRASP with path- relinking Numerical results Conclusions

3. February 2002GRASP with path-relinking for PVC routingSlide 3/42 (ROADEF) PVC routing: application 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

4. February 2002GRASP with path-relinking for PVC routingSlide 4/42 (ROADEF) PVC routing: example

5. February 2002GRASP with path-relinking for PVC routingSlide 5/42 (ROADEF) PVC routing: example

6. February 2002GRASP with path-relinking for PVC routingSlide 6/42 (ROADEF) PVC routing: example

7. February 2002GRASP with path-relinking for PVC routingSlide 7/42 (ROADEF) PVC routing: example

8. February 2002GRASP with path-relinking for PVC routingSlide 8/42 (ROADEF) PVC routing: example max capacity = 3

9. February 2002GRASP with path-relinking for PVC routingSlide 9/42 (ROADEF) PVC routing: example max capacity = 3very long path!

10. February 2002GRASP with path-relinking for PVC routingSlide 10/42 (ROADEF) PVC routing: example max capacity = 3very long path! reroute

11. February 2002GRASP with path-relinking for PVC routingSlide 11/42 (ROADEF) PVC routing: example max capacity = 3

12. February 2002GRASP with path-relinking for PVC routingSlide 12/42 (ROADEF) PVC routing: example max capacity = 3 feasible and optimal!

13. February 2002GRASP with path-relinking for PVC routingSlide 13/42 (ROADEF) PVC routing: application 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

14. February 2002GRASP with path-relinking for PVC routingSlide 14/42 (ROADEF) PVC routing: application Reorder PVCs and apply algorithm on switch to reroute: –taking advantage of factors not considered by switch algorithm may lead to greater network efficiency –FR switch algorithm is typically fast since it is also used to reroute in case of switch or trunk failures –this can be traded off for improved network resource utilization when routing off-line

15. February 2002GRASP with path-relinking for PVC routingSlide 15/42 (ROADEF) PVC routing: application 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

16. February 2002GRASP with path-relinking for PVC routingSlide 16/42 (ROADEF) Problem formulation 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

17. February 2002GRASP with path-relinking for PVC routingSlide 17/42 (ROADEF) Problem formulation 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

18. February 2002GRASP with path-relinking for PVC routingSlide 18/42 (ROADEF) Problem formulation 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)

19. February 2002GRASP with path-relinking for PVC routingSlide 19/42 (ROADEF) Problem formulation = 1, iff trunk (i,j ) is used to route PVC k.

20. February 2002GRASP with path-relinking for PVC routingSlide 20/42 (ROADEF) Cost function Linear combination of –delay component - weighted by (1-  ) –load balancing component - weighted by  Delay component:

21. February 2002GRASP with path-relinking for PVC routingSlide 21/42 (ROADEF) 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. February 2002GRASP with path-relinking for PVC routingSlide 22/42 (ROADEF) 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. February 2002GRASP with path-relinking for PVC routingSlide 23/42 (ROADEF) Some recent applications Laguna & Glover (1993): tabu search, different cost function, no constraints on PVCs routed on the same trunk (assign calls to paths) Sung & Park (1995): Lagrangean heuristic, very small graphs Amiri et al. (1999): Lagrangean heuristic, min delay Dahl et al. (1999): cutting planes (traffic assignment) Barnhart et al (2000): branch-and-price, different cost function, no constraints on PVCs routed on same trunk Shyur & Wen (2000): tabu search, min hubs

24. February 2002GRASP with path-relinking for PVC routingSlide 24/42 (ROADEF) Solution method: GRASP with path-relinking GRASP: Multistart metaheuristic, Feo & Resende 1989 Path-relinking: intensification, Glover (1996) Repeat for Max_Iterations: –Construct a greedy randomized solution –Use local search to improve the constructed solution –Apply path-relinking to further improve this solution –Update the pool of elite solutions –Update the best solution found

25. February 2002GRASP with path-relinking for PVC routingSlide 25/42 (ROADEF) Solution method: GRASP GRASP –Construction : RCL: n c unrouted PVCs with largest demands choose unrouted pair k biasing in favor of high bandwidth requirements, with probablity  k = r k / (  p  RCL r p ) capacity constraints relaxed and handled via the penalty function introduced by the load- balancing component length of each edge (i,j) is the incremental cost of routing r k additional units of demand on it route pair k using shortest route between its endpoints

26. February 2002GRASP with path-relinking for PVC routingSlide 26/42 (ROADEF) Solution method: GRASP GRASP –Local search: for each PVC k  K, remove r k units of flow from each edge in its current route recompute incremental weights of routing r k additional units of flow for all edges reroute PVC k using new shortest path

27. February 2002GRASP with path-relinking for PVC routingSlide 27/42 (ROADEF) Solution method: 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

28. February 2002GRASP with path-relinking for PVC routingSlide 28/42 (ROADEF) Solution method: 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

29. February 2002GRASP with path-relinking for PVC routingSlide 29/42 (ROADEF) 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) } Solution method: path- relinking

30. February 2002GRASP with path-relinking for PVC routingSlide 30/42 (ROADEF) Path-relinking in GRASP Maintain an elite set of solutions found during GRASP iterations. After each GRASP iteration (construction & local search): –Select an elite solution at random: guiding solution. –Use GRASP solution as initial solution. –Perform path-relinking between these two solutions.

31. February 2002GRASP with path-relinking for PVC routingSlide 31/42 (ROADEF) Path-relinking in GRASP Successful applications: –Prize-collecting Steiner tree problem Canuto, Resende, & Ribeiro (2000) –Steiner tree problem Ribeiro, Uchoa, & Werneck (2000) (e.g., best known results for open problems in series dv640 of the SteinLib) –Three-index assignment problem Aiex, Pardalos, Resende, & Toraldo (2000)

32. February 2002GRASP with path-relinking for PVC routingSlide 32/42 (ROADEF) Path-relinking: elite set P is 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.

33. February 2002GRASP with path-relinking for PVC routingSlide 33/42 (ROADEF)

34. February 2002GRASP with path-relinking for PVC routingSlide 34/42 (ROADEF) Experiment 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

35. February 2002GRASP with path-relinking for PVC routingSlide 35/42 (ROADEF) Experiment Test problems:

36. February 2002GRASP with path-relinking for PVC routingSlide 36/42 (ROADEF) Variants of path-relinking: –G: pure GRASP –GPRb: GRASP with backward PR –GPRf: GRASP with forward PR –GPRbf: GRASP with two-way PR Other strategies: –Truncated path-relinking –Do not apply PR at every iteration (frequency) Variants of GRASP and path- relinking S T T S S T S T

37. February 2002GRASP with path-relinking for PVC routingSlide 37/42 (ROADEF) Variants of GRASP and path- relinking Time (s) Probability Each variant: 200 runs for one instance of PVC routing problem

38. February 2002GRASP with path-relinking for PVC routingSlide 38/42 (ROADEF) 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

39. February 2002GRASP with path-relinking for PVC routingSlide 39/42 (ROADEF) 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.

40. February 2002GRASP with path-relinking for PVC routingSlide 40/42 (ROADEF) Parameter of the objective function Objective function  (solution) = Delay x (1-  ) + Load imbalance cost x  if  = 1: consider only trunk utilization rates if  = 0: consider only delays (capacities relaxed) increasing  0  1  minimization of maximum utilization rate dominates  reduction of flows in edges with higher loads  increase of flows in underloaded edges until the next breakpoint  flows concentrate around breakpoint levels  useful strategy for setting appropriate value of  to achieve some level of quality of service (max util.)

41. February 2002GRASP with path-relinking for PVC routingSlide 41/42 (ROADEF) Parameter of the objective function

42. February 2002GRASP with path-relinking for PVC routingSlide 42/42 (ROADEF) Concluding remarks (1/3) New formulation with flexible objective function Family of heuristics (greedy, greedy+LS, GRASP, GRASP+PR) Simple greedy heuristic improves algorithm in traffic engineering by network planners Objective function provides effective strategy for setting the weight parameter to achieve some quality of service level

43. February 2002GRASP with path-relinking for PVC routingSlide 43/42 (ROADEF) Concluding remarks (2/3) 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).

44. February 2002GRASP with path-relinking for PVC routingSlide 44/42 (ROADEF) Concluding remarks (3/3) 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.

45. February 2002GRASP with path-relinking for PVC routingSlide 45/42 (ROADEF) Slides and publications Slides of this talk can be downloaded from: http://www.inf.puc-rio/~celso/talks Paper about PVC routing available at: http://www.inf.puc- rio.br/~celso/publicacoes