September 2002 Parallel GRASP with PR for the 2-path network design problem 1/37 AIRO’2002 AIRO’2002 L’Aquila, September 10-13, 2002 A Parallel GRASP with Path-Relinking Heuristic for the 2-Path Network Design Problem Celso C. RIBEIRO Isabel ROSSETI Catholic University of Rio de Janeiro, Brazil

September 2002 Parallel GRASP with PR for the 2-path network design problem 2/37 AIRO’2002 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

September 2002 Parallel GRASP with PR for the 2-path network design problem 3/37 AIRO’ 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)

September 2002 Parallel GRASP with PR for the 2-path network design problem 4/37 AIRO’ 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

September 2002 Parallel GRASP with PR for the 2-path network design problem 5/37 AIRO’ 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

September 2002 Parallel GRASP with PR for the 2-path network design problem 6/37 AIRO’2002 GRASP with path-relinking GRASP: –Multistart metaheuristic: Feo & Resende (1989) Path-relinking: –Intensification strategy: 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 the solution –Update the pool of elite solutions –Update the best solution found

September 2002 Parallel GRASP with PR for the 2-path network design problem 7/37 AIRO’2002 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.

September 2002 Parallel GRASP with PR for the 2-path network design problem 8/37 AIRO’2002 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: 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.

September 2002 Parallel GRASP with PR for the 2-path network design problem 9/37 AIRO’2002 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 guiding solution path in the neighborhood of solutions

September 2002 Parallel GRASP with PR for the 2-path network design problem 10/37 AIRO’2002 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 selected: initial solution guiding solution

September 2002 Parallel GRASP with PR for the 2-path network design problem 11/37 AIRO’2002 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

September 2002 Parallel GRASP with PR for the 2-path network design problem 12/37 AIRO’2002 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.

September 2002 Parallel GRASP with PR for the 2-path network design problem 13/37 AIRO’2002 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 dv640 of the SteinLib) –Three-index assignment problem: Aiex et al. (2000) –Capacitated minimum spanning tree: Souza, Duhamel & Ribeiro (2002) (e.g., best known results for largest problems with 160 nodes)

September 2002 Parallel GRASP with PR for the 2-path network design problem 14/37 AIRO’2002 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.

September 2002 Parallel GRASP with PR for the 2-path network design problem 15/37 AIRO’2002

September 2002 Parallel GRASP with PR for the 2-path network design problem 16/37 AIRO’2002 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 –The processor acting as the master (data, seeds, iterations) also performs GRASP iterations –Each processor performs Max_Iterations/p iterations

September 2002 Parallel GRASP with PR for the 2-path network design problem 17/37 AIRO’2002 Parallel implementation: independent seed(1)seed(2)seed(3)seed(4)seed(p-1) Best solution is sent to the master p-1 Elite p seed(p)

September 2002 Parallel GRASP with PR for the 2-path network design problem 18/37 AIRO’2002 Parallel implementation 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) iterations

September 2002 Parallel GRASP with PR for the 2-path network design problem 19/37 AIRO’2002 Parallel implementation: cooperative 2 Elite 1 p 3 Elite solutions are stored in a centralized pool Master Slave

September 2002 Parallel GRASP with PR for the 2-path network design problem 20/37 AIRO’2002 Computational results Parallel GRASP heuristic: –Implementation in C –MPI LAM for communication –Linux cluster with 32 Pentium II-400 processors Largest instances: –Larger instances solved with the GRASP heuristic: |V|= 400, |E|= 79800, |D|= 4000 (previously: |V|= 120, |E|= 7140, |D|= 60)

September 2002 Parallel GRASP with PR for the 2-path network design problem 21/37 AIRO’2002 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. Parall el GRAS P Sample A D&J (2000 ) Sample B Size10030 Mean443.7 (- 2.2%) Std. dev

September 2002 Parallel GRASP with PR for the 2-path network design problem 22/37 AIRO’2002 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) Variants of GRASP with path- relinking S T T S S T S T

September 2002 Parallel GRASP with PR for the 2-path network design problem 23/37 AIRO’2002 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.

September 2002 Parallel GRASP with PR for the 2-path network design problem 24/37 AIRO’2002 Variants of GRASP with path- relinking Each variant: 200 runs for one instance of 2PNDP

September 2002 Parallel GRASP with PR for the 2-path network design problem 25/37 AIRO’2002 More recently: –G+PR(m): mixed back and forward strategy T: elite solution S: local search –Path-relinking with local search Variants of GRASP with path- relinking T S

September 2002 Parallel GRASP with PR for the 2-path network design problem 26/37 AIRO’2002 Variants of GRASP with path- relinking Each variant: 200 runs for one instance of 2PNDP Probabilit y Time (seconds)

September 2002 Parallel GRASP with PR for the 2-path network design problem 27/37 AIRO’2002 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 Strategies using the backwards component are systematically better

September 2002 Parallel GRASP with PR for the 2-path network design problem 28/37 AIRO’2002 Independent strategy: speedups Linear speedups: |V|= 400, 3200 iterations, G+PR(BF)

September 2002 Parallel GRASP with PR for the 2-path network design problem 29/37 AIRO’2002 Cooperative vs. independent strategy Solution quality Same instance: 15 runs with different seeds, 3200 iterations The pool is poorer when fewer GRASP iterations are performed Proc s. bestavg.bestavg Independent Cooperative

September 2002 Parallel GRASP with PR for the 2-path network design problem 30/37 AIRO’2002 Cooperative vs. independent strategy Procs.Indep. Coop

September 2002 Parallel GRASP with PR for the 2-path network design problem 31/37 AIRO’2002 Cooperative vs. independent strategy Select an instance and a target value. For each strategy: –Perform 100 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.

September 2002 Parallel GRASP with PR for the 2-path network design problem 32/37 AIRO’2002 Probability Cooperative vs. independent strategy 2 processors Cooperative Independent Time to target value (seconds)

September 2002 Parallel GRASP with PR for the 2-path network design problem 33/37 AIRO’2002 Probability Cooperative vs. independent strategy 4 processors Cooperative Independent Time to target value (seconds)

September 2002 Parallel GRASP with PR for the 2-path network design problem 34/37 AIRO’2002 Probability Cooperative vs. independent strategy 8 processors Cooperative Independent Time to target value (seconds)

September 2002 Parallel GRASP with PR for the 2-path network design problem 35/37 AIRO’2002 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.

September 2002 Parallel GRASP with PR for the 2-path network design problem 36/37 AIRO’2002 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. Robust cooperative strategy is faster and better.

September 2002 Parallel GRASP with PR for the 2-path network design problem 37/37 AIRO’2002 Slides and publications Slides of this talk can be downloaded from: rio/~celso/talks Paper about the parallel GRASP heuristic for the 2-path network design problem available at: rio.br/~celso/publicacoes Isabel Rosseti