1/37 AIRO

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AIRO2002 LAquila, 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 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)

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2-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

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2-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

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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
• Set the modified weights equal to the original
  weights.
• Randomly select an origin-destination pair (a,b)
  ? D.
• Compute a shortest 2-path between a and b using
  the modified weights.
• Set to 0 the modified weights of the edges in
  this path.
• Remove (a,b) from D.
• If D is empty stop, otherwise go back to step 2.

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GRASP with path-relinking

• GRASP
• Local search phase
• Generate a circular random permutation of the
  pairs in D.
• Select the next origin-destination pair (a,b) ?
  D.
• 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.
• If D paths have been investigated without
  improvement stop, otherwise go back to step 2.

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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.

guiding solution
path in the neighborhood of solutions
initial solution
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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

guiding solution
initial solution
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GRASP with path-relinking

• Elite solutions x and y
• ?(x,y) symmetric difference between x and y
• while ( ?(x,y) gt 0 )
• evaluate moves corresponding in ?(x,y) make
  best move
• update ?(x,y)

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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.

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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 treeSouza, Duhamel
  Ribeiro (2002) (e.g., best known results for
  largest problems with 160 nodes)

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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.

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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)
Elite
Elite
Elite
Elite
Elite
1
2
3
4
p-1
Best solution is sent to the master
Elite
p
seed(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
Master
Elite solutions are stored in a centralized pool
Elite
1
p
2
3
Slave
Slave
Slave
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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 DJ (2000) Sample 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

• Variants of GRASP with path-relinking
• GRASP pure GRASP
• GPR(B) GRASP with backward PR
• GPR(F) GRASP with forward PR
• GPR(BF) GRASP with two-way PRT elite solution
  S local search
• Other strategies
• Truncated path-relinking
• Do not apply PR at every iteration (frequency)

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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.

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Variants of GRASP with path-relinking
Each variant 200 runs for one instance of 2PNDP
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Variants of GRASP with path-relinking

• More recently
• GPR(m) mixed back and forward strategyT elite
  solution S local search
• Path-relinking with local search

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Variants of GRASP with path-relinking
Each variant 200 runs for one instance of 2PNDP
Probability
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 ? GPR(F) ? GPR(B) ?
  GPR(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(GPR(F),10s)
  56P(GPR(B),10s) 75 P(GPR(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,
  GPR(BF)

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Cooperative vs. independent strategy
Independent
Cooperative

• 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
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Cooperative vs. independent strategy
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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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.

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Cooperative vs. independent strategy
2 processors
Cooperative Independent
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Cooperative vs. independent strategy
4 processors
Cooperative Independent
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Cooperative vs. independent strategy
8 processors
Cooperative Independent
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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.

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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