4 èmes journées de la roadef paris, february 20-22, 2002

February 2002

Parallel GRASP for the 2-path network design problem

Slide 1/25 (ROADEF)

4èmes Journées de la ROADEFParis, February 20-22, 2002

A Parallel GRASP Heuristic for the 2-Path Network Design Problem

Celso C. RIBEIRO Isabel ROSSETI

Catholic University of Rio de Janeiro Brazil

Metro Corvisart, Paris 13ème

February 2002


Slide 2/25 (ROADEF)

Summary• Problem formulation• GRASP with path-relinking heuristic

– Construction phase– Local search phase– Path-relinking

• Parallel implementation• Computational results• Concluding remarks

February 2002


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2-path network design problem

• Graph G = (V,E)V: node setE: edge setweights 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)

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Slide 4/25 (ROADEF)


• 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

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• 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

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Slide 6/25 (ROADEF)

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

February 2002


Slide 7/25 (ROADEF)

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.

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Slide 8/25 (ROADEF)

GRASP with path-relinking• GRASP

– Local search phase1. 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.

February 2002


Slide 9/25 (ROADEF)


• 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

guidingsolution

path in neighborhood of solutions

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Slide 10/25 (ROADEF)

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:

Initialsolution

guiding solution

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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)}


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• 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.

February 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)

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• 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.

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Parallel implementation• Main interest of parallel implementations of

metaheuristics: robustnessCung, Martins, Ribeiro & Roucairol (2001)

• Parallelization strategy: – Multiple-walk independent-thread strategy– Iterations evenly distributed over p processors– Each processor keeps a copy of the algorithm and

data– One processor acts as the master (data, seeds,

iterations)– Each processor performs Max_Iterations/p

iterations

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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)

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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.

Parallel

GRASP

Sample A

D&J (2000

) Sampl

e B

Size 100 30

Mean 443.7 (-

2.2%)

453.7

Std. dev.

40.6 61.6

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• 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

• Other strategies:– Truncated path-relinking– Do not apply PR at every iteration

(frequency)

Variants of GRASP with path-relinking

S T

TS

S T

S T

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• 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.

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Each variant: 200 runs for one instance of 2PNDP

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.1 1.0 10.0 100.0 1000.0 10000.0

Pro

ba

bili

ty

Time (seconds)

G+PR(BF)G+PR(B)G+PR(F)GRASP

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• 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

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Speedups• Linear speedups: |V|= 400, 3200 iterations

1.02.0

4.0

7.9

15.7

31.2

1 2 4 8 16 32

Sp

ee

du

p

Processors

GRASP+PR

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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.

February 2002



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(after March 15, 2002)

Isabel Rosseti

4 èmes journées de la roadef paris, february 20-22, 2002

Documents

constructed solutionapply

edge e ekpath

meparallel grasp

weights of edges

parallel grasp heuristic

d paths

small delays parallel

original weights