PAREO 2002 Guadeloupe May 20 24 2002 A

PAREO 2002 Guadeloupe, May 20 -24, 2002 A Parallel GRASP with Path-Relinking Heuristic for the 2 -Path Network Design Problem Celso C. RIBEIRO Isabel ROSSETI Gosier, Guadeloupe May 2002 1 Catholic University of Rio de Janeiro Brazil Parallel GRASP with PR for the 2 -path network design problem

Summary • Problem formulation • GRASP with path-relinking heuristic – Construction phase – Local search phase – Path-relinking • Parallel implementation – Independent strategy – Cooperative strategy • Computational results • Concluding remarks May 2002 2 Parallel GRASP with PR for the 2 -path network design problem

2 -path network design problem • Graph G = (V, E) V: node set E: edge set weights we 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) May 2002 3 Parallel GRASP with PR for the 2 -path network design problem

2 -path network design problem • 2 -path network design problem (2 PNDP): 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 May 2002 4 Parallel GRASP with PR for the 2 -path network design problem

2 -path network design problem • Dahl & Johannessen (2000): – Decision version of 2 PNDP is NP-complete. – Approximate algorithm – Exact cutting plane algorithm • Balakrishnan & Altinkemer (1992): – Integer programming formulation for k. PNDP – See also Le. Blanc, Chifflet & Mahey (1999). • Generalizations: k-hop minimum spanning tree, k-hop minimum Steiner tree May 2002 5 Parallel GRASP with PR for the 2 -path network design problem

GRASP with path-relinking • GRASP: – Multistart metaheuristic: Feo & Resende (1989) • Path-relinking: – Intensification strategy: Glover (1996) • Repeat for Max_Iterations: – – – May 2002 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 6 Parallel GRASP with PR for the 2 -path network design problem

GRASP with path-relinking • 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. May 2002 7 Parallel GRASP with PR for the 2 -path network design problem

GRASP with path-relinking • 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: • • May 2002 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| 8 paths have been investigated without Parallel GRASP with PR for the 2 -path network design improvement stop, otherwise goproblem back to step 2.

GRASP with 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 May 2002 path in neighborhood of solutions 9 guiding solution Parallel GRASP with PR for the 2 -path network design problem

GRASP with 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 May 2002 10 Parallel GRASP with PR for the 2 -path network design problem

GRASP with 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) } May 2002 11 Parallel GRASP with PR for the 2 -path network design problem

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. 12 May 2002 Parallel GRASP with PR for the 2 -path network design problem

GRASP with path-relinking • Successful applications: – Prize-collecting Steiner tree problem: Canuto, Resende & Ribeiro (2001) – Minimum Steiner tree problem: Ribeiro, Uchoa & Werneck (2002) (e. g. , best known results for open problems in series dv 640 of the Stein. Lib) – Three-index assignment problem: Aiex et al. (2000) – Capacitated minimum spanning tree: Souza, Duhamel & Ribeiro (2002) (e. g. , best 13 May 2002 Parallel GRASP with PR for the 2 -path network design known results for largest problems problemwith 160

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 in Parallel P. GRASP with PR for the 2 -path network design 14 May 2002 from all solutions problem

May 2002 15 Parallel GRASP with PR for the 2 -path network design problem

Parallel implementation • Main interest of parallel implementations of metaheuristics: robustness Cung, Martins, Ribeiro & Roucairol (2001) • Multiple-walk independent-thread strategy: – p processors available – Iterations evenly distributed over the p processors – Each processor keeps a copy of the algorithm and data – One processor acts as the master (data, seeds, iterations), but also performs GRASP iterations – Each processor performs Max_Iterations/p iterations May 2002 16 Parallel GRASP with PR for the 2 -path network design problem

Parallel implementation: independent seed(1) seed(2) Elite 1 seed(3) Elite 3 2 p seed(p) 17 Elite 4 Elite May 2002 seed(4) seed(p-1) Elite p-1 Best solution is sent to the master Parallel GRASP with PR for the 2 -path network design problem

Parallel implementation • Main interest of parallel implementations of metaheuristics: robustness Cung, Martins, Ribeiro & Roucairol (2001) • Multiple-walk cooperative-thread strategy: – p processors available – Iterations evenly distributed over p-1 processors – Each processor keeps a copy of the algorithm and data – One processor acts as the master (data, seeds, iterations) and controls the pool of elite solutions – Each slave processor performs Max_Iterations/(p-1) May 2002 Parallel GRASP with PR for the 2 -path network design iterations 18 problem

Parallel implementation: cooperative Master Elite solutions are stored in a centralized pool Elite 1 2 Slave May 2002 p 3 19 Slave Parallel GRASP with PR for the 2 -path network design problem

Computational results • 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) May 2002 20 Parallel GRASP with PR for the 2 -path network design problem

Computational results • 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. May 2002 21 Parallel D&J GRASP (2000) Size Sample A Sample B 100 30 Mean 443. 7 ( 453. 7 -2. 2%) Std. dev. 40. 6 61. 6 Parallel GRASP with PR for the 2 -path network design problem

Variants of GRASP with path-relinking • Variants of GRASP with path-relinking: – – T S GRASP: pure GRASP G+PR(B): GRASP with backward PR S G+PR(F): GRASP with forward PR G+PR(BF): GRASP with two-way PRS T: elite solution S: local search • Other strategies: S – Truncated path-relinking – Do not apply PR at every iteration (frequency) May 2002 22 Parallel GRASP with PR for the 2 -path network design problem T T T

Variants of GRASP with pathrelinking • Select an instance and a target value. • For each variant of GRASP with pathrelinking: – 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 time. Parallel GRASP with PR for the 2 -path network design 23 May 2002 computation problem

Variants of GRASP with pathrelinking Each variant: 200 runs for one instance of 2 PNDP May 2002 24 Parallel GRASP with PR for the 2 -path network design problem

Variants of GRASP with pathrelinking • Same computation time: probability of finding a solution at least as good as the target value increases from GRASP G+PR(F) 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, 10 s) = 2% P(G+PR(F), 10 s) = 56% P(G+PR(B), 10 s) = 75% P(G+PR(BF), 10 s) = 84% • Effectiveness of path-relinking to improve and 25 May 2002 Parallel GRASP with PR for the 2 -path network design problem speedup the pure GRASP

Independent strategy: speedups • Linear speedups: |V|= 400, 3200 iterations, G+PR(BF) May 2002 26 Parallel GRASP with PR for the 2 -path network design problem

Cooperative vs. independent strategy Independent Cooperative • Solution quality Procs. best avg. • Same instance: 1 520 525. 4 15 runs with 2 519 524. 5 519 526. 4 different seeds, 3200 iterations 4 527. 8 521 526. 3 • The pool is poorer 8 524 529. 5 521 526. 5 when fewer 533 535. 1 515 525. 0 GRASP iterations 16 are performed 32 538 541. 2 521 526. 3 May 2002 27 Parallel GRASP with PR for the 2 -path network design problem

Cooperative vs. independent strategy Procs. Indep. Coop. 1 May 2002 28 1358. 5 - 2 682. 2 2192. 1 4 333. 0 740. 4 8 165. 0 312. 4 16 81. 6 197. 9 32 41. 2 182. 7 Parallel GRASP with PR for the 2 -path network design problem

Cooperative vs. independent strategy • Select an instance and a target value. • For each strategy: – Perform 100 runs using different seeds (not all runs already performed). – 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. 29 May 2002 Parallel GRASP with PR for the 2 -path network design problem

Cooperative vs. independent strategy May 2002 30 Parallel GRASP with PR for the 2 -path network design problem

Cooperative vs. independent strategy May 2002 31 Parallel GRASP with PR for the 2 -path network design problem

Cooperative vs. independent strategy May 2002 32 Parallel GRASP with PR for the 2 -path network design problem

Cooperative vs. independent strategy • Recall that when p processors are used: – All of them perform GRASP iterations in the independent strategy – Only p-1 processors perform GRASP iterations in the cooperative strategy • Cooperative strategy improves w. r. t. the independent strategy when the number of processors increases. • Cooperative strategy is already faster for p 4 processors. May 2002 33 Parallel GRASP with PR for the 2 -path network design problem

Concluding remarks • New heuristic for the 2 -path network design problem. • Effectiveness of the new heuristic: – Larger problems solved. – New heuristic finds better solutions. – Domination is stronger for harder or larger instances. • 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). • Linear speedups with the parallel implementation. 34 May 2002 GRASP with PR forbetter. the 2 -path network design • Cooperative strategy is. Parallel faster and problem

Slides and publications • Slides of this talk can be downloaded from: http: //www. inf. puc-rio/~celso/talks • Paper about the parallel GRASP heuristic for the 2 -path network design problem available at: http: //www. inf. puc-rio. br/~celso/publicacoes Isabel Rosseti May 2002 35 Parallel GRASP with PR for the 2 -path network design problem