1. September 2003 GRASP and path-relinking: Advances and applications 1/82 CEMAPRE GRASP and Path- Relinking: Advances and Applications Celso C. RIBEIRO 7th CEMAPRE CONFERENCE Lisbon, September 18-19, 2003

2. September 2003 GRASP and path-relinking: Advances and applications 2/82 CEMAPRE Summary The need for heuristics Basic GRASP algorithm Construction phase Enhanced construction strategies Local search Path-relinking GRASP with path-relinking Variants of GRASP with path-relinking Concluding remarks... + Applications and numerical results

3. September 2003 GRASP and path-relinking: Advances and applications 3/82 CEMAPRE The need for heuristics Decision problem: is there a solution satisfying some given conditions? Answer: yes or no Optimization problem: among all solutions satisfying some given conditions, find one optimizing a cost function. Answer: a feasible optimal solution Complexity theory: decision problems

4. September 2003 GRASP and path-relinking: Advances and applications 4/82 CEMAPRE The need for heuristics Example: traveling salesman problem Input: n cities and distances c ij Decision problem: given an integer L, is there a hamiltonian cycle of length less than or equal to L? Optimization problem: find a minimum length hamiltonian cycle.

5. September 2003 GRASP and path-relinking: Advances and applications 5/82 CEMAPRE The need for heuristics Class P: decision problems solvable by polynomial time algorithms. Examples: shortest paths, minimum spanning trees, maximum flow, linear programming, matching, etc. Class NP: decision problems for which one can check in polynomial time if a solution leads to a yes answer (but not necessarilly find one).

6. September 2003 GRASP and path-relinking: Advances and applications 6/82 CEMAPRE The need for heuristics NP-complete problems: hardest decision problems in NP, if there exists a polynomial time algorithm for any of them, then all of them can be solved in polynomial time. NP-hard problems: optimization problems whose associated decision problem is NP-complete. Examples: traveling salesman problem, knapsack problem, graph coloring, Steiner problem in graphs, integer programming, and many others.

7. September 2003 GRASP and path-relinking: Advances and applications 7/82 CEMAPRE How do we cope with NP-hardness? –Heuristics: approximate methods tailored for each problem, which are able to find high quality feasible solutions in reasonable computation times. –Strategies: The need for heuristics Local search algorithms: improve feasible solutions and stop at first local optimum Metaheuristics: go beyond local optimality Construction algorithms: build feasible solutions

8. September 2003 GRASP and path-relinking: Advances and applications 8/82 CEMAPRE The need for heuristics Metaheuristics are based on different metaphors: –Simulated annealing –Tabu search –Genetic algorithms –Scatter search –GRASP (greedy randomized adaptive search procedures) –Variable neighborhood search –Ant colonies –Swarm particles –Asynchronous teams (A-teams), …

9. September 2003 GRASP and path-relinking: Advances and applications 9/82 CEMAPRE The need for heuristics Effective implementation of metaheuristics combine several common ingredients: –Greedy algorithms, randomization, local search, intensification, diversification, short-term memory, learning, elite solutions, etc. Advances in heuristic search methods: –Solve larger problems –Solve problems in smaller times –Find better solutions

10. September 2003 GRASP and path-relinking: Advances and applications 10/82 CEMAPRE Summary The need for heuristics Basic GRASP algorithm Construction phase Enhanced construction strategies Local search Path-relinking GRASP with path-relinking Variants of GRASP with path- relinking Concluding remarks

11. September 2003 GRASP and path-relinking: Advances and applications 11/82 CEMAPRE GRASP: –Multistart metaheuristic: Feo & Resende (1989): set covering Festa & Resende (2002): annotated bibliography Resende & Ribeiro (2003): survey Repeat for Max_Iterations: –Construct a greedy randomized solution. –Use local search to improve the constructed solution. –Update the best solution found. GRASP: Basic algorithm

12. September 2003 GRASP and path-relinking: Advances and applications 12/82 CEMAPRE Construction phase: greediness + randomness –Builds a feasible solution: Use greediness to build restricted candidate list and apply randomness to select an element from the list. Use randomness to build restricted candidate list and apply greediness to select an element from the list. Local search: search for an improving neighbor solution until a local optimum is found –Solutions generated by the construction procedure are not necessarily optimal: Effectiveness of local search depends on neighborhood structure, search strategy, and fast evaluation of neighbors, but also on the construction procedure itself. GRASP: Basic algorithm

13. September 2003 GRASP and path-relinking: Advances and applications 13/82 CEMAPRE weight iterations random construction local search GRASP: Basic algorithm Application: modem placement max weighted covering problem maximization problem: = 0.85 iterations GRASP construction local search weight Effectiveness of greedy randomized vs purely randomized construction:

14. September 2003 GRASP and path-relinking: Advances and applications 14/82 CEMAPRE Summary The need for heuristics Basic GRASP algorithm Construction phase Enhanced construction strategies Local search Path-relinking GRASP with path-relinking Variants of GRASP with path- relinking Parallel implementations Concluding remarks

15. September 2003 GRASP and path-relinking: Advances and applications 15/82 CEMAPRE Greedy Randomized Construction: –Solution –Evaluate incremental costs of candidate elements –While Solution is not complete do: Build restricted candidate list (RCL). Select an element s from RCL at random. Solution Solution {s} Reevaluate the incremental costs. –endwhile Construction phase

16. September 2003 GRASP and path-relinking: Advances and applications 16/82 CEMAPRE Construction phase Minimization problem Basic construction procedure: –Greedy function c(e): incremental cost associated with the incorporation of element e into the current partial solution under construction –c min (resp. c max ): smallest (resp. largest) incremental cost –RCL made up by the elements with the smallest incremental costs.

17. September 2003 GRASP and path-relinking: Advances and applications 17/82 CEMAPRE Construction phase Cardinality-based construction: –p elements with the smallest incremental costs Quality-based construction: –Parameter defines the quality of the elements in RCL. –RCL contains elements with incremental cost c min c(e) c min + (c max –c min ) – = 0 : pure greedy construction – = 1 : pure randomized construction Select at random from RCL using uniform probability distribution

18. September 2003 GRASP and path-relinking: Advances and applications 18/82 CEMAPRE best solution average solution time time (seconds) for 1000 iterations solution value RCL parameter α Illustrative results: RCL parameter randomgreedy weighted MAX-SAT instance: 100 variables and 850 clauses 1000 GRASP iterations SGI Challenge 196 MHz

19. September 2003 GRASP and path-relinking: Advances and applications 19/82 CEMAPRE Illustrative results: RCL parameter Another weighted MAX-SAT instance randomgreedy RCL parameter α SGI Challenge 196 MHz

20. September 2003 GRASP and path-relinking: Advances and applications 20/82 CEMAPRE Summary The need for heuristics Basic GRASP algorithm Construction phase Enhanced construction strategies Local search Path-relinking GRASP with path-relinking Variants of GRASP with path- relinking Concluding remarks

21. September 2003 GRASP and path-relinking: Advances and applications 21/82 CEMAPRE Enhanced construction strategies Reactive GRASP: Prais & Ribeiro (2000) (traffic assignment in TDMA satellites) –At each GRASP iteration, a value of the RCL parameter is chosen from a discrete set of values [ 1, 2,..., m ]. –The probability that k is selected is p k. –Reactive GRASP: adaptively changes the probabilities [p 1, p 2,..., p m ] to favor values of that produce good solutions. –Other applications, e.g. to graph planarization, set covering, and weighted max-sat: –Better solutions, at the cost of slightly larger times.

22. September 2003 GRASP and path-relinking: Advances and applications 22/82 CEMAPRE Enhanced construction strategies Cost perturbations: Canuto, Resende, & Ribeiro (2001) (prize-collecting Steiner tree) –Randomly perturb original costs and apply some heuristic. –Adds flexibility to algorithm design: May be more effective than greedy randomized construction in circumstances where the construction algorithm is not very sensitive to randomization. Also useful when no greedy algorithm is available.

23. September 2003 GRASP and path-relinking: Advances and applications 23/82 CEMAPRE Enhanced construction strategies Memory and learning in construction: Fleurent & Glover (1999) (quadratic assignment) –Uses long-term memory (pool of elite solutions) to favor elements which frequently appear in the elite solutions (consistent and strongly determined variables).

24. September 2003 GRASP and path-relinking: Advances and applications 24/82 CEMAPRE Summary The need for heuristics Basic GRASP algorithm Construction phase Enhanced construction strategies Local search Path-relinking GRASP with path-relinking Variants of GRASP with path- relinking Concluding remarks

25. September 2003 GRASP and path-relinking: Advances and applications 25/82 CEMAPRE Local search First improving vs. best improving: –First improving is usually faster. –Premature convergence to low quality local optimum is more likely to occur with best improving. VND to speedup search and to overcome optimality w.r.t. to simple neighborhoods: Ribeiro, Uchoa, & Werneck (2002) (Steiner problem in graphs) Hashing to avoid cycling or repeated application of local search to same solution built in the construction phase: Woodruff & Zemel (1993), Ribeiro et. al (1997) (query optimization), Martins et al. (2000) (Steiner problem in graphs)

26. September 2003 GRASP and path-relinking: Advances and applications 26/82 CEMAPRE Local search Filtering to avoid application of local search to low quality solutions, only promising solutions are investigated: Feo, Resende, & Smith (1994), Prais & Ribeiro (2000) (traffic assignment), Martins et. al (2000) (Steiner problem in graphs) Extended quick-tabu local search to overcome premature convergence: Souza, Duhamel, & Ribeiro (2003) (capacitated minimum spanning tree, better solutions for largest benchmark problems) Candidate list strategies

27. September 2003 GRASP and path-relinking: Advances and applications 27/82 CEMAPRE Summary The need for heuristics Basic GRASP algorithm Construction phase Enhanced construction strategies Local search Path-relinking GRASP with path-relinking Variants of GRASP with path- relinking Concluding remarks

28. September 2003 GRASP and path-relinking: Advances and applications 28/82 CEMAPRE Path-relinking Path-relinking: –Intensification strategy exploring trajectories connecting elite solutions: Glover (1996) –Originally proposed in the context of tabu search and scatter search. –Paths in the solution space leading to other elite solutions are explored in the search for better solutions: selection of moves that introduce attributes of a guiding solution into the current solution

29. September 2003 GRASP and path-relinking: Advances and applications 29/82 CEMAPRE Path-relinking Exploration of trajectories that connect high quality (elite) solutions: initial solution guiding solution path in the neighborhood of solutions

30. September 2003 GRASP and path-relinking: Advances and applications 30/82 CEMAPRE 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 selected: initial solution guiding solution

31. September 2003 GRASP and path-relinking: Advances and applications 31/82 CEMAPRE 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

32. September 2003 GRASP and path-relinking: Advances and applications 32/82 CEMAPRE Summary The need for heuristics Basic GRASP algorithm Construction phase Enhanced construction strategies Local search Path-relinking GRASP with path-relinking Variants of GRASP with path- relinking Concluding remarks

33. September 2003 GRASP and path-relinking: Advances and applications 33/82 CEMAPRE GRASP with path-relinking Maintains a set of elite solutions found during GRASP iterations. After each GRASP iteration (construction and local search): –Use GRASP solution as initial solution. –Select an elite solution uniformly at random: guiding solution (may also be selected with probabilities proportional to the symmetric difference w.r.t. the initial solution). –Perform path-relinking between these two solutions.

34. September 2003 GRASP and path-relinking: Advances and applications 34/82 CEMAPRE GRASP with path-relinking Repeat for Max_Iterations: –Construct a greedy randomized solution. –Use local search to improve the constructed solution. –Apply path-relinking to further improve the solution. –Update the pool of elite solutions. –Update the best solution found.

35. September 2003 GRASP and path-relinking: Advances and applications 35/82 CEMAPRE GRASP with path-relinking Variants: trade-offs between computation time and solution quality –Explore different trajectories (e.g. backward, forward): better start from the best, neighborhood of the initial solution is fully explored! –Explore both trajectories: twice as much the time, often with marginal improvements only! –Do not apply PR at every iteration, but instead only periodically: similar to filtering during local search. –Truncate the search, do not follow the full trajectory. –May also be applied as a post-optimization step to all pairs of elite solutions.

36. September 2003 GRASP and path-relinking: Advances and applications 36/82 CEMAPRE GRASP with path-relinking Successful applications: 1)Prize-collecting minimum Steiner tree problem: Canuto, Resende, & Ribeiro (2001) (e.g. improved all solutions found by approximation algorithm of Goemans & Williamson) 2)Minimum Steiner tree problem: Ribeiro, Uchoa, & Werneck (2002) (e.g., best known results for open problems in series dv640 of the SteinLib) 3)p-median: Resende & Werneck (2002) (e.g., best known solutions for problems in literature)

37. September 2003 GRASP and path-relinking: Advances and applications 37/82 CEMAPRE GRASP with path-relinking Successful applications (contd): 4)Capacitated minimum spanning tree: Souza, Duhamel, & Ribeiro (2002) (e.g., best known results for largest problems with 160 nodes) 5)2-path network design: Ribeiro & Rosseti (2002) (better solutions than greedy heuristic) 6)Max-cut: Festa, Pardalos, Resende, & Ribeiro (2002) (e.g., best known results for several instances) 7)Quadratic assignment: Oliveira, Pardalos, & Resende (2003)

38. September 2003 GRASP and path-relinking: Advances and applications 38/82 CEMAPRE GRASP with path-relinking Successful applications (contd): 8)Job-shop scheduling: Aiex, Binato, & Resende (2003) 9)Three-index assignment problem: Aiex, Resende, Pardalos, & Toraldo (2003) 10)PVC routing: Resende & Ribeiro (2003) 11)Phylogenetic trees: Ribeiro & Vianna (2003)

39. September 2003 GRASP and path-relinking: Advances and applications 39/82 CEMAPRE GRASP with path-relinking P is a set (pool) 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.

40. September 2003 GRASP and path-relinking: Advances and applications 40/82 CEMAPRE Local access network design Design a local access network taking into account tradeoff between: –cost of network –revenue potential of network

41. September 2003 GRASP and path-relinking: Advances and applications 41/82 CEMAPRE Local access network design Build an optical fyber network to provide large bandwidth services to commercial and residential users, taking into account the construction costs (fyber laying) and the potential revenues obtained by providing the services to the selected clients (loss of revenue in the case of clients which are not served). Prize-collecting Steiner tree problem: application developped for AT&T

42. September 2003 GRASP and path-relinking: Advances and applications 42/82 CEMAPRE Local access network design residenc e (revenue ) Street (cost)

43. September 2003 GRASP and path-relinking: Advances and applications 43/82 CEMAPRE potential equipment location (cost) coverage area of equipment

44. September 2003 GRASP and path-relinking: Advances and applications 44/82 CEMAPRE area of coverage potential equipment location Locate p boxes to maximize revenue of covered residences.

45. September 2003 GRASP and path-relinking: Advances and applications 45/82 CEMAPRE equipment box Assign to each equipment box the total revenue of residences it covers for which there is no closer box. This is the prize.

46. September 2003 GRASP and path-relinking: Advances and applications 46/82 CEMAPRE Solve prize-collecting Steiner tree problem Maximize prize collected minus edge costs: Here all prizes are collected.

47. September 2003 GRASP and path-relinking: Advances and applications 47/82 CEMAPRE Solve prize collecting Steiner tree problem Maximize prize collected minus edge costs: Here not all prizes are collected.

48. September 2003 GRASP and path-relinking: Advances and applications 48/82 CEMAPRE Prize collecting Steiner tree problem Typical dimension: 20,000 to 100,000 nodes. Use GRASP for maximum covering to locate equipment boxes. Compute lower bounds with cutting planes algorithm of Lucena & Resende (2000). Compute solutions (upper bounds) with GRASP (local search with perturbations) algorithm of Canuto, Resende, & Ribeiro (2001).

49. September 2003 GRASP and path-relinking: Advances and applications 49/82 CEMAPRE Prize collecting Steiner tree problem Construction phase (each iteration): –Memory-based perturbation of original costs to diversify previously obtained solutions. –Determine nodes to be collected using Goemans & Williamson (1996) 2-opt primal-dual algorithm. –Connect the selected nodes by a minimum spanning tree. Local search: add/drop nodes to/from the current tree. PR: nodes in one solution but not in the other

50. September 2003 GRASP and path-relinking: Advances and applications 50/82 CEMAPRE Prize collecting Steiner tree problem Computational results: –114 test problems (up to 1,000 nodes and 25,000 edges) –Heuristic found: 96 out of 104 known optimal solutions. Solution within 1% of lower bound for 104 of 114 problems. –PR more than doubled the number of optima found. –Cutting plane algorithm produces tight lower bounds but running times are very high for the largest instances (days, even weeks), while the GRASP heuristic averaged less than 3 hours for the same instances.

51. September 2003 GRASP and path-relinking: Advances and applications 51/82 CEMAPRE Summary The need for heuristics Basic GRASP algorithm Construction phase Enhanced construction strategies Local search Path-relinking GRASP with path-relinking Variants of GRASP with path- relinking Concluding remarks

52. September 2003 GRASP and path-relinking: Advances and applications 52/82 CEMAPRE Probability distribution of time-to- target-solution-value: –Aiex, Resende, & Ribeiro (2002) and Aiex, Binato, & Resende (2003) have shown experimentally that for both pure GRASP and GRASP with path-relinking the time- to-target-solution-value follows a two- parameter exponential distribution. –Consequence: linear speedups for independent parallel implementations Time-to-target-value plots

53. September 2003 GRASP and path-relinking: Advances and applications 53/82 CEMAPRE Probability distribution of time-to- target-solution-value: experimental plots 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. Time-to-target-value plots

54. September 2003 GRASP and path-relinking: Advances and applications 54/82 CEMAPRE Random variable time-to-target-solution value fits a two-parameter exponential distribution. Therefore, one should expect approximate linear speedup in a straightforward (independent) parallel implementation. Time-to-target-value plots

55. September 2003 GRASP and path-relinking: Advances and applications 55/82 CEMAPRE 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 T: elite solution S: local search Other strategies: –Truncated path-relinking –Do not apply PR at every iteration (frequency) S T T S S T S T Variants of GRASP + PR

56. September 2003 GRASP and path-relinking: Advances and applications 56/82 CEMAPRE 2-path network design problem: –Graph G=(V,E) with edge weights w e and set D of origin-destination pairs (demands): find a minimum weighted subset of edges E E containing a 2-path (path with at most two edges) 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 2-path network design problem

57. September 2003 GRASP and path-relinking: Advances and applications 57/82 CEMAPRE 2-path network design problem Each variant: 200 runs for one instance of 2PNDP Sun Sparc Ultra 1 80 nodes, 800 pairs, target=5 88

58. September 2003 GRASP and path-relinking: Advances and applications 58/82 CEMAPRE 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% 2-path network design problem

59. September 2003 GRASP and path-relinking: Advances and applications 59/82 CEMAPRE More recently: –G+PR(M): mixed back and forward strategy T: elite solution S: local search –Path-relinking with local search –Randomized path-relinking T S Variants of GRASP + PR

60. September 2003 GRASP and path-relinking: Advances and applications 60/82 CEMAPRE 2-path network design problem Each variant: 200 runs for one instance of 2PNDP Sun Sparc Ultra 1 80 nodes, 800 pairs, target=5 88

61. September 2003 GRASP and path-relinking: Advances and applications 61/82 CEMAPRE Instanc e GRAS P G+PR( F) G+PR(B ) G+PR(F B) G+PR( M) 100-3773762756757754 100-5756742739737728 200-315641523151615081509 200-515771567154315291531 300-324482381233923562338 300-524502364232823472322 400-333883311326832273257 400-534163335326732703259 500-343474239418741704187 500-543624263420342114200 10 runs, same computatio n time for each variant, best solution found 2-path network design problem

62. September 2003 GRASP and path-relinking: Advances and applications 62/82 CEMAPRE Effectiveness of G+PR(M): –100 small instances with 70 nodes generated as in Dahl and Johannessen (2000) for comparison purposes. –Statistical test t for unpaired observations –GRASP finds better solutions with 40% of confidence (unpaired observations and many optimal solutions): G+PR( M) Sample A D&J Sample B Size10030 Mean443.7 (-2.2%) 453.7 Std. dev. 40.661.6 2-path network design problem Ribeiro & Rosseti (2002)

63. September 2003 GRASP and path-relinking: Advances and applications 63/82 CEMAPRE Effectiveness of path-relinking to improve and speedup the pure GRASP. Strategies using the backwards component are systematically better. 2-path network design problem

64. September 2003 GRASP and path-relinking: Advances and applications 64/82 CEMAPRE PVC routing Frame relay service offers virtual private networks to customers by providing long- term private virtual circuits (PVCs) between customer endpoints on a backbone network. Routing is done either automatically by switch or by the network designer without any knowledge of future requests. Over time, these decisions cause inefficiencies in the network and occasionally offline rerouting (grooming) of the PVCs is needed: –integer multicommodity network flow problem: Resende & Ribeiro (2003)

65. September 2003 GRASP and path-relinking: Advances and applications 65/82 CEMAPRE PVC routing

66. September 2003 GRASP and path-relinking: Advances and applications 66/82 CEMAPRE PVC routing

67. September 2003 GRASP and path-relinking: Advances and applications 67/82 CEMAPRE PVC routing

68. September 2003 GRASP and path-relinking: Advances and applications 68/82 CEMAPRE PVC routing

69. September 2003 GRASP and path-relinking: Advances and applications 69/82 CEMAPRE PVC routing max capacity = 3

70. September 2003 GRASP and path-relinking: Advances and applications 70/82 CEMAPRE PVC routing max capacity = 3very long path!

71. September 2003 GRASP and path-relinking: Advances and applications 71/82 CEMAPRE PVC routing max capacity = 3very long path! reroute

72. September 2003 GRASP and path-relinking: Advances and applications 72/82 CEMAPRE PVC routing max capacity = 3

73. September 2003 GRASP and path-relinking: Advances and applications 73/82 CEMAPRE PVC routing max capacity = 3 feasible and optimal!

74. September 2003 GRASP and path-relinking: Advances and applications 74/82 CEMAPRE time (seconds) Probability Each variant: 200 runs for one instance of PVC routing problem (60 nodes, 498 edges, 750 origin-destination pairs) PVC routing SGI Challenge 196 MHz

75. September 2003 GRASP and path-relinking: Advances and applications 75/82 CEMAPRE PVC routing 10 runs 10 seconds100 seconds Variantbest avera ge best avera ge GRASP 126603126695126228126558 G+PR(F) 126301126578126083126229 G+PR(B) 125960126281125666125883 G+PR(BF ) 125961126307125646125850

76. September 2003 GRASP and path-relinking: Advances and applications 76/82 CEMAPRE PVC routing 10 runs 10 seconds100 seconds Variantbest avera ge best avera ge GRASP 126603126695126228126558 G+PR(F) 126301126578126083126229 G+PR(B) 125960126281125666125883 G+PR(BF ) 125961126307125646125850

77. September 2003 GRASP and path-relinking: Advances and applications 77/82 CEMAPRE PVC routing GRASP + PR backwards: four increasingly difficult target values Same behavior, plots drift to the right for more difficult targets SGI Challenge 196 MHz

78. September 2003 GRASP and path-relinking: Advances and applications 78/82 CEMAPRE Post-optimization evolutionary strategy : a)Start with pool P 0 found at end of GRASP and set k = 0. b)Combine with path-relinking all pairs of solutions in P k. c)Solutions obtained by combining solutions in P k are added to a new pool P k+1 following same constraints for updates as before. d)If best solution of P k+1 is better than best solution of P k, then set k = k + 1, and go back to step (b). Succesfully used by Ribeiro, Uchoa, & Werneck (2002) (Steiner) and Resende & Werneck (2002) (p-median) GRASP with path-relinking

79. September 2003 GRASP and path-relinking: Advances and applications 79/82 CEMAPRE Summary The need for heuristics Basic GRASP algorithm Construction phase Enhanced construction strategies Local search Path-relinking GRASP with path-relinking Variants of GRASP with path- relinking Concluding remarks

80. September 2003 GRASP and path-relinking: Advances and applications 80/82 CEMAPRE Concluding remarks (1/2) Path-relinking adds memory and intensification mechanisms to GRASP, systematically contributing to improve solution quality: –better solutions in smaller times –some implementation strategies appear to be more effective than others). –mixed path-relinking strategy is very promising –backward relinking is usually more effective than forward –bidirectional relinking does not necessarily pays the additional computation time

81. September 2003 GRASP and path-relinking: Advances and applications 81/82 CEMAPRE Concluding remarks (2/2) Difficulties: –How to deal with infeasibilities along the relinking procedure? –How to apply path-relinking in partitioning problems such as graph- coloring, bin packing, and others? Other applications of path-relinking: –VNS+PR: Festa, Pardalos, Resende, & Ribeiro (2002) –PR as a generalized optimized crossover in genetic algorithms: Ribeiro & Vianna (2003) Effective parallel strategies based on path-relinking to introduce memory and processor cooperation (robustness and linear speedups).

82. September 2003 GRASP and path-relinking: Advances and applications 82/82 CEMAPRE Slides, publications, and acknowledgements Slides of this talk can be downloaded from: http://www.inf.puc-rio/~celso/talks Papers about GRASP, path-relinking, and their applications available at: http://www.inf.puc-rio.br/~celso/publicacoes Joint work done mostly with Maurício Resende (AT&T Labs) and M.Sc. and Ph.D. students from PUC-Rio, who are all gratefully acknowledged: S. Canuto, M. Souza, M. Prais, S. Martins, D. Vianna, R. Aiex, R. Werneck, E. Uchoa, and I. Rosseti.