Slide 1/59 (EHESS)

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Recent Developments in Metaheuristics
and Applications to Network Design Problems
Celso C. RIBEIRO
Catholic University of Rio de
Janeiro, Brazil Department of Computer Science
Joint work with Maurício Resende and Isabel
Rosseti
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Summary

• 2-path network design problem
• PVC routing problem
• GRASP with path-relinking
• Construction phase
• Local search phase
• Extended local search
• Path-relinking
• Parallel implementation
• Numerical results
• Variants of path-relinking
• Effectiveness of the new heuristics
• Parallelization
• 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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PVC routing

• Virtual private networks permanent virtual
  circuits (PVCs) between customer endpoints on a
  backbone network
• Routing either automatically by switch or by
  network designer without any knowledge of future
  requests
• Inefficiencies and occasional need for off-line
  rerouting of the PVCs

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PVC routing example
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PVC routing example
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PVC routing example
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PVC routing example
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PVC routing example
max capacity 3
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PVC routing example
max capacity 3
reroute
very long path!
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PVC routing example
max capacity 3
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PVC routing example
max capacity 3
feasible and optimal!
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PVC routing

• Other algorithms simply handle
  the number of hops (e.g. routing
  algorithm in Cisco switches)
• Handling delays is particularly important in
  international networks, where distances between
  backbone nodes vary considerably

Cisco Catalystic 5505 switch
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PVC routing

• Load balancing is important for providing
  flexibility to handle
• overbooking typically used by network designers
  to account for non-coincidence of traffic
• PVC rerouting due to failures
• bursting above the committed rate not only
  allowed, but also sold to customers as one of the
  attractive features of frame relay
• Integer multicommodity network flow problem

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PVC routing

• Given undirected FR network G (V, E), where
• V denotes n backbone nodes (FR switches)
• E denotes m trunks connecting backbone nodes
• for each trunk e (i,j )
• b (e ) maximum bandwidth (max kbits/sec rate)
• c (e ) maximum number of PVCs that can be routed
  on it
• d (e ) propagation and hopping delay

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PVC routing

• Demands K 1,,p defined by
• Origin-destination pairs (o,d )
• r (p) effective bandwidth requirement (forward,
  backward, overbooking) for PVC p
• Objective is to minimize
• delays
• network load unbalance
• subject to
• technological constraints

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PVC routing

• route for PVC (o,d ) is a sequence of adjacent
  trunks from node o to node d
• set of routing assignments is feasible if for all
  trunks e
• total bandwidth requirements routed on e does
  exceed b (e)
• number of PVCs routed on e not greater than c(e)

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Cost function

• Linear combination of
• delay component - weighted by (1-?)
• load balancing component - weighted by ?
• Delay component

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Cost function

• Load balancing component measure of Fortz
  Thorup (2000) to compute congestion
• ? ?1(L1) ?2(L2)
  ?E(LE)
• where Le is the load on link e ? E,
• ?e(Le) is piecewise linear and
  convex,
• ?e(0) 0, for all e ? E.

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Piecewise linear and convex ?e(Le) link
congestion measure
(Le?ce)
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GRASP with path-relinking

• GRASP
• Multistart metaheuristic, Feo Resende (1989)
• Path-relinking
• Intensification strategy, Glover (1996)
• Repeat for MaxIterations
• Construct greedy randomized solution
• Use local search to improve constructed solution
• Apply path-relinking to further improve solution
• Update pool of elite solutions
• Update best solution found

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Greedy Randomized Adaptive Search Procedures
(GRASP)

• Combinaison dune méthode constructive avec une
  approche par recherche locale, dans une procédure
  itérative (multi-départ) où les itérations sont
  totalement indépendantes les unes des autres
• (a) construction dune solution
• (b) recherche locale

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Greedy Randomized Adaptive Search Procedures
(GRASP)

• f(s) ? ?
• for i 1,,MaxIterations do
• Construire une solution s en utilisant un
  algorithme glouton randomisé
• Appliquer une procédure de recherche locale
  à partir de s, pour obtenir la solution s
• if f(s) lt f(s) then s ? s
• end-for

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GRASP

• Phase de construction construire une solution
  réalisable, un élément à la fois
• Chaque itération
• évaluer le bénéfice de chaque élément en
  utilisant une fonction gloutonne
• créer une liste restricte de candidats, formée
  par les éléments avec les meilleures évaluations
• sélectionner de façon probabiliste un élément de
  la liste restricte de candidats
• adapter la fonction gloutonne après lutilisation
  de lélément choisi

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GRASP

• Restriction des éléments dans la liste de
  candidats
• nombre maximum déléments dans la liste
• qualité des éléments dans la liste (par rapport
  au choix exclusivement glouton)cmin bénefice
  minimum parmi tous les élémentscmax bénefice
  maximum parmi tous les éléments
• Paramètre a permet de controler la qualité des
  éléments dans la liste de candidats Liste
  éléments j cj cmin a . (cmax - cmin) a
  0 choix glouton a 1 choix probabiliste

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GRASP

• Choix probabiliste entre les meilleurs éléments
  de la liste de candidats (pas forcément le
  meilleur, comme cest le cas du choix
  exclusivement glouton)
• la qualité moyenne de la solution dépend de la
  qualité des éléments dans la liste
• la diversité des solutions construites dépend du
  nombre déléments dans la liste
• Diversification basée sur la randomisation
  controlée différentes solutions construites lors
  de différentes iterations GRASP

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GRASP

• Construction gloutonne bonnes solutions (près
  des optima locaux), permettant daccélérer la
  recherche locale
• Recherche locale amélioration des solutions
  construites lors de la première phase
• choix du voisinage
• structures de données efficaces pour accélérer la
  recherche locale
• bonnes solutions de départ permettent daccélérer
  la recherche locale

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GRASP

• Technique déchantillonage
  de lespace de
  recherche

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GRASP

• Implémentation simple
  algorithme glouton recherche locale
• Peu de paramètres à régler
• restrictivité de la liste de candidats
• nombre ditérations
• Desavantage nutilise pas de mémoire
• Parallélisation simple et directe
• N itérations, p processeurs chaque processeur
  fait N/p itérations et à la fin informe au maître
  la meilleure solution quil a trouvée

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GRASP with extended local search

• Basic GRASP procedure puts too much effort in the
  construction phase, but stops in the first local
  optimum.
• Strategy to increase the effort in the local
  search phase and to add memory to GRASP.
• Improved local search by using a very short-term
  memory (tabu search).

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GRASP with extended local search

• Keep a tabu list with the last TabuTenure (small
  value, typically 5-10) solutions visited.
• Move to the best neighbor solution (be it an
  improving solution or not) which is not in the
  tabu list.
• Stop after TabuTenure iterations without
  improvement in the best global solution.
• Best results for capacitated minimum spanning
  tree Souza, Duhamel Ribeiro (2002)

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GRASP for 2-path network design

• 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 for 2-path network design

• 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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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 neighborhood of solutions
initial solution
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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
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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.
• PR allows introducing memory into GRASP.

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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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Path-relinking with GRASP

• Successful applications
• Prize-collecting Steiner tree problem
  Canuto, Resende Ribeiro (2001)
• 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, Pardalos, Resende Toraldo (2000)
• Capacitated minimum spanning treeSouza, Duhamel
  Ribeiro (2002) (e.g., best known results for
  largest problems with 160 nodes)

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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 results2-path network design

• 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 results2-path network design

• 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 PR
• 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

• 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

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Computational results PVC routing

• Heuristics
• H1 sorts demands in decreasing order and routes
  them using minimum hops paths
• H2 sorts demands in decreasing order and routes
  using same cost function as GRASP
• H3 adds the same local search to H2
• GPRb GRASP with backwards path-relinking
• SGI Challenge 196 MHz

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Computational results PVC routing

• Test problems

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Variants of GRASP and path-relinking
Each variant 200 runs for one instance of PVC
routing problem
Probability
Time (s)
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Variants of GRASP and path-relinking

• Same computation time probability of finding a
  solution at least as good as the target value
  increases from G ? GPRf ? GPRfb ? GPRb
• P(h,t) probability variant h finds solution as
  good as target value in time no greater than t
• P(GPRfb,100s)9.25 P(GPRb,100s)28.75
• P(G,2000s)8.33 P(GPRf,2000s)65.25
• P(h,time)50 Times for each variant
• GPRb129s G10933s GPRf1727s GPRfb172s

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Comparisons
cost
max util.
Distribution 86/60/2 86 edges with utilization
in 0,1/3), 60 in 1/3,2/3),
and two in 2/3,9/10)
In general GPRB gt H3 gt H2 gt H1 (cost, max
utilization, distribution)
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Parallel implementation speedups

• Linear speedups 2PNDP, V 400, 3200 iterations

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Concluding remarks (1/3)

• Effectiveness of the new heuristic for the 2-path
  network design problem
• Larger problems solved.
• New heuristic finds better solutions.
• Domination is stronger for harder or
  larger instances.
• Heuristic for PVC routing improves algorithm used
  in traffic engineering by network planners.

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Concluding remarks (2/3)

• NETROUTER Tool for optimally loading demands on
  single-path routes on a capacitated network. It
  uses the GPRb variant of the combination of GRASP
  and path-relinking, minimizing delays while
  balancing network load.
• Application - Netrouter is currently being used
  for the design of ATT's next generation
  frame-relay and MPLS core architecture, to assess
  if the current and forecasted demands can be
  handled by the proposed trunking plan.

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Concluding remarks (3/3)

• 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 (e.g., backwards from
  better, elite solution to current locally optimal
  solution).
• Extended local search also adds memory and
  improves the solutions obtained by basic local
  search.
• Linear speedups with the parallel mplementation.

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Slides and publications

• Slides of this talk can be downloaded from
  http//www.inf.puc-rio/celso/talks
• Papers about
• recent developments and state-of-the-art on
  GRASP
• GRASP with path-relinking heuristic for PVC
  routing
• parallel GRASP heuristic for the 2-path network
  design problem and
• applications of GRASP and path-relinking
• on which this talk was based are available at
• http//www.inf.puc-rio.br/celso/publicacoes