airo’2002 l’aquila, september 10-13, 2002

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

1/37 AIRO’2002

AIRO’2002L’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


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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


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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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• 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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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


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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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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.


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• 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 the neighborhood of solutions


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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:

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.


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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)

• 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


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Parallel implementation: independent

seed(1) seed(2) seed(3) seed(4) seed(p-1)

Best solution is sent to the master

1 2 3 4 p-1Elite Elite Elite Elite Elite

Elite

pseed(p)


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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


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Parallel implementation: cooperative

2

Elite

1

p3

Elite solutions are stored in a centralized pool Master

Slave SlaveSlave


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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: – 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

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

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

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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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• More recently:– G+PR(m): mixed back and forward

strategyT: elite solution S: local search

– Path-relinking with local search


TS


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0.0

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prob

abili

dade

tempo para valor alvo

GGPRfGPRb

GPRfbGPRm


Each variant: 200 runs for one instance of 2PNDPPro

bab

ilit

y

Time (seconds)


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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


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Independent strategy: speedups• Linear speedups: |V|= 400, 3200 iterations, G+PR(BF)

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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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

Procs.

best

avg. best avg.

1 520 525.4

- -

2 519 524.5

519 526.4

4 524 527.8

521 526.3

8 524 529.5

521 526.5

16 533 535.1

515 525.0

32 538 541.2

521 526.3

Independent Cooperative


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0

500

1000

1500

2000

1 2 4 8 16 32

Tim

e (

s)

Processors

independentcooperative


Procs.

Indep.

Coop.

1 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


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• 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.


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abili

dade


CooperativoIndependente

Pro

bab

ility


2 processors

CooperativeIndependent

Time to target value (seconds)


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4 processors




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8 processors




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• 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.


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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.

• Robust cooperative strategy is faster and better.


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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

airo’2002 l’aquila, september 10-13, 2002

Documents

constructed solutionapply

edge e ekpath

weights of edges

brazilparallel grasp

d paths

original weights

small delays parallel

original edge weights