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A GRASP with path-relinking heuristic for PVC routing

A GRASP with path-relinking heuristic for PVC routing. Celso C. Ribeiro Computer Science Department Catholic University of Rio de Janeiro j oint work with M.G.C. Resende. Summary . PVC routing Integer multicommodity flow formulation Cost function

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A GRASP with path-relinking heuristic for PVC routing

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  1. 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 GRASP with path-relinking for PVC routing

  2. Summary • PVC routing • Integer multicommodity flow formulation • Cost function • Solution method: GRASP with path-relinking • Numerical results • Conclusions GRASP with path-relinking for PVC routing

  3. PVC routing: application • Virtual private networks: permanent virtualcircuits (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 GRASP with path-relinking for PVC routing

  4. PVC routing: example GRASP with path-relinking for PVC routing

  5. PVC routing: example GRASP with path-relinking for PVC routing

  6. PVC routing: example GRASP with path-relinking for PVC routing

  7. PVC routing: example GRASP with path-relinking for PVC routing

  8. PVC routing: example max capacity = 3 GRASP with path-relinking for PVC routing

  9. PVC routing: example very long path! max capacity = 3 GRASP with path-relinking for PVC routing

  10. PVC routing: example very long path! max capacity = 3 reroute GRASP with path-relinking for PVC routing

  11. PVC routing: example max capacity = 3 GRASP with path-relinking for PVC routing

  12. PVC routing: example max capacity = 3 feasible and optimal! GRASP with path-relinking for PVC routing

  13. PVC routing: application • Other algorithms simply handle the number of hops (e.g. routing algorithmin Cisco switches) • Handling delays is particularly important in international networks,where distances between backbone nodes vary considerably GRASP with path-relinking for PVC routing

  14. 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 GRASP with path-relinking for PVC routing

  15. PVC routing: application • Load balancing is important for providing flexibility to handle: • overbooking: typically used by network designers to account fornon-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 GRASP with path-relinking for PVC routing

  16. 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 GRASP with path-relinking for PVC routing

  17. Problem formulation • DemandsK = {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 GRASP with path-relinking for PVC routing

  18. 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) GRASP with path-relinking for PVC routing

  19. = 1, iff trunk (i,j ) is used to route PVC k. Problem formulation GRASP with path-relinking for PVC routing

  20. Cost function • Linear combination of • delay component - weighted by (1-) • load balancing component - weighted by  • Delay component: GRASP with path-relinking for PVC routing

  21. Cost function • Load balancing component: measure of Fortz & Thorup (2000) to compute congestion:  = 1(L1) + 2(L2) + … + |E|(L|E|) where Leis the load on link e  E, e(Le) is piecewise linear and convex, e(0) = 0, for all e  E. GRASP with path-relinking for PVC routing

  22. slope = 5000 slope = 500 slope = 3 slope = 10 slope = 1 slope = 70 Piecewise linear and convex e(Le) link congestion measure (Lece) GRASP with path-relinking for PVC routing

  23. 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 GRASP with path-relinking for PVC routing

  24. 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 GRASP with path-relinking for PVC routing

  25. Solution method: GRASP • GRASP • Construction: • RCL: nc unrouted PVCs with largest demands • choose unrouted pair kbiasing in favor of high bandwidth requirements, with probablity k = rk / (pRCLrp ) • 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 rk additional units of demand on it • route pair k using shortest route between its endpoints GRASP with path-relinking for PVC routing

  26. Solution method: GRASP • GRASP • Local search: • for each PVC kK , remove rk units of flow from each edge in its current route • recompute incremental weights of routing rk additional units of flow for all edges • reroute PVC k using new shortest path GRASP with path-relinking for PVC routing

  27. 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. guiding solution path in neighborhood of solutions initial solution GRASP with path-relinking for PVC routing

  28. 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: guiding solution Initial solution GRASP with path-relinking for PVC routing

  29. Solution method: path-relinking 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) } GRASP with path-relinking for PVC routing

  30. 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. GRASP with path-relinking for PVC routing

  31. 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) GRASP with path-relinking for PVC routing

  32. 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 . GRASP with path-relinking for PVC routing

  33. GRASP with path-relinking for PVC routing

  34. 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 GRASP with path-relinking for PVC routing

  35. Experiment • Test problems: GRASP with path-relinking for PVC routing

  36. T S T S S T S T Variants of GRASP and path-relinking • 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) GRASP with path-relinking for PVC routing

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

  38. 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 GRASP with path-relinking for PVC routing

  39. cost max util. 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) GRASP with path-relinking for PVC routing

  40. 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.) GRASP with path-relinking for PVC routing

  41. Parameter of the objective function GRASP with path-relinking for PVC routing

  42. 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 GRASP with path-relinking for PVC routing

  43. 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). GRASP with path-relinking for PVC routing

  44. Concluding remarks (3/3) • NETROUTER – Tool for optimallyloading 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 thecurrent and forecasted demands can be handled by the proposed trunkingplan. GRASP with path-relinking for PVC routing

  45. 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 GRASP with path-relinking for PVC routing

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