Post on 15-Mar-2016
description
September 2003 GRASP and path-relinking: Advances and applications
1/82 CEMAPRE
GRASP and Path-Relinking:
Advances and Applications
Celso C. RIBEIRO
7th CEMAPRE CONFERENCE
Lisbon, September 18-19, 2003
September 2003 GRASP and path-relinking: Advances and applications
2/82 CEMAPRE
Summary• The need for heuristics• Basic GRASP algorithm• Construction phase• Enhanced construction strategies• Local search• Path-relinking• GRASP with path-relinking• Variants of GRASP with path-relinking• Concluding remarks... + Applications and numerical results
September 2003 GRASP and path-relinking: Advances and applications
3/82 CEMAPRE
The need for heuristics • Decision problem: “is there a solution
satisfying some given conditions?” Answer: “yes” or “no”
• Optimization problem: “among all solutions satisfying some given conditions, find one optimizing a cost function.” Answer: a feasible optimal solution
• Complexity theory: decision problems
September 2003 GRASP and path-relinking: Advances and applications
4/82 CEMAPRE
The need for heuristics • Example: traveling salesman problem
Input: n cities and distances cij
Decision problem: “given an integer L, is there a hamiltonian cycle of length less than or equal to L?”
Optimization problem: “find a minimum length hamiltonian cycle.”
September 2003 GRASP and path-relinking: Advances and applications
5/82 CEMAPRE
The need for heuristics• Class P: decision problems solvable
by polynomial time algorithms. • Examples: shortest paths, minimum
spanning trees, maximum flow, linear programming, matching, etc.
• Class NP: decision problems for which one can check in polynomial time if a solution leads to a “yes” answer (but not necessarilly find one).
September 2003 GRASP and path-relinking: Advances and applications
6/82 CEMAPRE
The need for heuristics• NP-complete problems: hardest decision
problems in NP, if there exists a polynomial time algorithm for any of them, then all of them can be solved in polynomial time.
• NP-hard problems: optimization problems whose associated decision problem is NP-complete.• Examples: traveling salesman problem, knapsack problem, graph coloring, Steiner problem in graphs, integer programming, and many others.
September 2003 GRASP and path-relinking: Advances and applications
7/82 CEMAPRE
• How do we cope with NP-hardness?– Heuristics: approximate methods tailored for
each problem, which are able to find high quality feasible solutions in “reasonable” computation times.
– Strategies:
The need for heuristics
•Local search algorithms: improve feasible solutions and stop at first local optimum
•Metaheuristics: go beyond local optimality
•Construction algorithms: build feasible solutions
September 2003 GRASP and path-relinking: Advances and applications
8/82 CEMAPRE
The need for heuristics• Metaheuristics are based on different
metaphors:– Simulated annealing– Tabu search– Genetic algorithms– Scatter search– GRASP (greedy randomized adaptive search
procedures)– Variable neighborhood search– Ant colonies– Swarm particles– Asynchronous teams (A-teams), …
September 2003 GRASP and path-relinking: Advances and applications
9/82 CEMAPRE
The need for heuristics• Effective implementation of
metaheuristics combine several common ingredients:– Greedy algorithms, randomization, local
search, intensification, diversification, short-term memory, learning, elite solutions, etc.
• Advances in heuristic search methods:– Solve larger problems– Solve problems in smaller times– Find better solutions
September 2003 GRASP and path-relinking: Advances and applications
10/82 CEMAPRE
Summary• The need for heuristics• Basic GRASP algorithm• Construction phase• Enhanced construction strategies• Local search• Path-relinking• GRASP with path-relinking• Variants of GRASP with path-relinking• Concluding remarks
September 2003 GRASP and path-relinking: Advances and applications
11/82 CEMAPRE
• GRASP: – Multistart metaheuristic:
• Feo & Resende (1989): set covering• Festa & Resende (2002): annotated bibliography• Resende & Ribeiro (2003): survey
• Repeat for Max_Iterations:– Construct a greedy randomized solution.– Use local search to improve the constructed
solution.– Update the best solution found.
GRASP: Basic algorithm
September 2003 GRASP and path-relinking: Advances and applications
12/82 CEMAPRE
• Construction phase: greediness + randomness– Builds a feasible solution:
• Use greediness to build restricted candidate list and apply randomness to select an element from the list.
• Use randomness to build restricted candidate list and apply greediness to select an element from the list.
• Local search: search for an improving neighbor solution until a local optimum is found– Solutions generated by the construction
procedure are not necessarily optimal:• Effectiveness of local search depends on neighborhood
structure, search strategy, and fast evaluation of neighbors, but also on the construction procedure itself.
GRASP: Basic algorithm
September 2003 GRASP and path-relinking: Advances and applications
13/82 CEMAPRE
weig
ht
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
0 500 1000 1500 2000 2500 3000 3500 4000
phase 2 solnphase 1 soln
weig
ht
iterations
random construction
local search
GRASP: Basic algorithm
Application: modem placement max weighted covering problemmaximization problem: = 0.85
9.509.559.609.659.709.759.809.859.909.95
10.00
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
t)
phase 2 solnphase 1 soln
iterations
GRASP construction
local search
weig
htEffectiveness of greedy randomized vs
purely randomized construction:
September 2003 GRASP and path-relinking: Advances and applications
14/82 CEMAPRE
Summary• The need for heuristics• Basic GRASP algorithm• Construction phase• Enhanced construction strategies• Local search• Path-relinking• GRASP with path-relinking• Variants of GRASP with path-
relinking• Parallel implementations• Concluding remarks
September 2003 GRASP and path-relinking: Advances and applications
15/82 CEMAPRE
• Greedy Randomized Construction:– Solution – Evaluate incremental costs of candidate
elements– While Solution is not complete do:
•Build restricted candidate list (RCL).•Select an element s from RCL at random.•Solution Solution {s}•Reevaluate the incremental costs.
– endwhile
Construction phase
September 2003 GRASP and path-relinking: Advances and applications
16/82 CEMAPRE
Construction phase• Minimization problem• Basic construction procedure:
– Greedy function c(e): incremental cost associated with the incorporation of element e into the current partial solution under construction
– cmin (resp. cmax): smallest (resp. largest) incremental cost
– RCL made up by the elements with the smallest incremental costs.
September 2003 GRASP and path-relinking: Advances and applications
17/82 CEMAPRE
Construction phase• Cardinality-based construction:
– p elements with the smallest incremental costs• Quality-based construction:
– Parameter defines the quality of the elements in RCL.
– RCL contains elements with incremental cost cmin c(e) cmin + (cmax –cmin)
= 0 : pure greedy construction = 1 : pure randomized construction
• Select at random from RCL using uniform probability distribution
September 2003 GRASP and path-relinking: Advances and applications
18/82 CEMAPRE
0
5
10
15
20
25
30
35
40
45
0 0.2 0.4 0.6 0.8 1 400000
405000
410000
415000
420000
425000
430000
435000
440000
445000
450000
best solution
average solution
time
time
(sec
onds
) for
100
0 ite
ratio
ns
solu
tion
valu
e
RCL parameter α
Illustrative results: RCL parameter
random greedy
weighted MAX-SAT instance: 100 variables and 850 clauses1000 GRASP iterations SGI Challenge 196 MHz
September 2003 GRASP and path-relinking: Advances and applications
19/82 CEMAPRE
5
10
15
20
0 0.2 0.4 0.6 0.8 1
time
(sec
onds
) for
100
0 ite
ratio
ns
RCL parameter alpha
total CPU time
local search CPU time
Illustrative results: RCL parameter
Another weighted MAX-SAT instance
random greedyRCL parameter αSGI Challenge 196 MHz
September 2003 GRASP and path-relinking: Advances and applications
20/82 CEMAPRE
Summary• The need for heuristics• Basic GRASP algorithm• Construction phase• Enhanced construction strategies• Local search• Path-relinking• GRASP with path-relinking• Variants of GRASP with path-relinking• Concluding remarks
September 2003 GRASP and path-relinking: Advances and applications
21/82 CEMAPRE
Enhanced construction strategies• Reactive GRASP: Prais & Ribeiro (2000)
(traffic assignment in TDMA satellites)– At each GRASP iteration, a value of the RCL
parameter is chosen from a discrete set of values [1, 2, ..., m].
– The probability that k is selected is pk.– Reactive GRASP: adaptively changes the
probabilities [p1, p2, ..., pm] to favor values of that produce good solutions.
– Other applications, e.g. to graph planarization, set covering, and weighted max-sat:
– Better solutions, at the cost of slightly larger times.
September 2003 GRASP and path-relinking: Advances and applications
22/82 CEMAPRE
Enhanced construction strategies
• Cost perturbations: Canuto, Resende, & Ribeiro (2001) (prize-collecting Steiner tree)– Randomly perturb original costs and apply
some heuristic.– Adds flexibility to algorithm design:
• May be more effective than greedy randomized construction in circumstances where the construction algorithm is not very sensitive to randomization.
• Also useful when no greedy algorithm is available.
September 2003 GRASP and path-relinking: Advances and applications
23/82 CEMAPRE
Enhanced construction strategies
• Memory and learning in construction: Fleurent & Glover (1999) (quadratic assignment)– Uses long-term memory (pool of elite
solutions) to favor elements which frequently appear in the elite solutions (consistent and strongly determined variables).
September 2003 GRASP and path-relinking: Advances and applications
24/82 CEMAPRE
Summary• The need for heuristics• Basic GRASP algorithm• Construction phase• Enhanced construction strategies• Local search• Path-relinking• GRASP with path-relinking• Variants of GRASP with path-relinking• Concluding remarks
September 2003 GRASP and path-relinking: Advances and applications
25/82 CEMAPRE
Local search• First improving vs. best improving:
– First improving is usually faster.– Premature convergence to low quality local optimum
is more likely to occur with best improving.• VND to speedup search and to overcome
optimality w.r.t. to simple neighborhoods: Ribeiro, Uchoa, & Werneck (2002) (Steiner problem in graphs)
• Hashing to avoid cycling or repeated application of local search to same solution built in the construction phase: Woodruff & Zemel (1993), Ribeiro et. al (1997) (query optimization), Martins et al. (2000) (Steiner problem in graphs)
September 2003 GRASP and path-relinking: Advances and applications
26/82 CEMAPRE
Local search• Filtering to avoid application of local
search to low quality solutions, only promising solutions are investigated: Feo, Resende, & Smith (1994), Prais & Ribeiro (2000) (traffic assignment), Martins et. al (2000) (Steiner problem in graphs)
• Extended quick-tabu local search to overcome premature convergence: Souza, Duhamel, & Ribeiro (2003) (capacitated minimum spanning tree, better solutions for largest benchmark problems)
• Candidate list strategies
September 2003 GRASP and path-relinking: Advances and applications
27/82 CEMAPRE
Summary• The need for heuristics• Basic GRASP algorithm• Construction phase• Enhanced construction strategies• Local search• Path-relinking• GRASP with path-relinking• Variants of GRASP with path-relinking• Concluding remarks
September 2003 GRASP and path-relinking: Advances and applications
28/82 CEMAPRE
Path-relinking• Path-relinking:
– Intensification strategy exploring trajectories connecting elite solutions: Glover (1996)
– Originally proposed in the context of tabu search and scatter search.
– Paths in the solution space leading to other elite solutions are explored in the search for better solutions:• selection of moves that introduce attributes
of a guiding solution into the current solution
September 2003 GRASP and path-relinking: Advances and applications
29/82 CEMAPRE
Path-relinking• Exploration of trajectories that
connect high quality (elite) solutions:initial
solutionguidingsolution
path in the neighborhood of solutions
September 2003 GRASP and path-relinking: Advances and applications
30/82 CEMAPRE
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
solutionguiding solution
September 2003 GRASP and path-relinking: Advances and applications
31/82 CEMAPRE
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)}
Path-relinking
September 2003 GRASP and path-relinking: Advances and applications
32/82 CEMAPRE
Summary• The need for heuristics• Basic GRASP algorithm• Construction phase• Enhanced construction strategies• Local search• Path-relinking• GRASP with path-relinking• Variants of GRASP with path-relinking• Concluding remarks
September 2003 GRASP and path-relinking: Advances and applications
33/82 CEMAPRE
GRASP with path-relinking• Maintains a set of elite solutions found
during GRASP iterations.• After each GRASP iteration (construction
and local search):– Use GRASP solution as initial solution. – Select an elite solution uniformly at random:
guiding solution (may also be selected with probabilities proportional to the symmetric difference w.r.t. the initial solution).
– Perform path-relinking between these two solutions.
September 2003 GRASP and path-relinking: Advances and applications
34/82 CEMAPRE
GRASP with path-relinking• 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 2003 GRASP and path-relinking: Advances and applications
35/82 CEMAPRE
GRASP with path-relinking• Variants: trade-offs between computation
time and solution quality– Explore different trajectories (e.g. backward,
forward): better start from the best, neighborhood of the initial solution is fully explored!
– Explore both trajectories: twice as much the time, often with marginal improvements only!
– Do not apply PR at every iteration, but instead only periodically: similar to filtering during local search.
– Truncate the search, do not follow the full trajectory.
– May also be applied as a post-optimization step to all pairs of elite solutions.
September 2003 GRASP and path-relinking: Advances and applications
36/82 CEMAPRE
GRASP with path-relinking• Successful applications:
1) Prize-collecting minimum Steiner tree problem: Canuto, Resende, & Ribeiro (2001) (e.g. improved all solutions found by approximation algorithm of Goemans & Williamson)
2) Minimum Steiner tree problem: Ribeiro, Uchoa, & Werneck (2002) (e.g., best known results for open problems in series dv640 of the SteinLib)
3) p-median: Resende & Werneck (2002) (e.g., best known solutions for problems in literature)
September 2003 GRASP and path-relinking: Advances and applications
37/82 CEMAPRE
GRASP with path-relinking• Successful applications (cont’d):
4) Capacitated minimum spanning tree:Souza, Duhamel, & Ribeiro (2002) (e.g., best known results for largest problems with 160 nodes)
5) 2-path network design: Ribeiro & Rosseti (2002) (better solutions than greedy heuristic)
6) Max-cut: Festa, Pardalos, Resende, & Ribeiro (2002) (e.g., best known results for several instances)
7) Quadratic assignment: Oliveira, Pardalos, & Resende (2003)
September 2003 GRASP and path-relinking: Advances and applications
38/82 CEMAPRE
GRASP with path-relinking• Successful applications (cont’d):
8) Job-shop scheduling: Aiex, Binato, & Resende (2003)
9) Three-index assignment problem: Aiex, Resende, Pardalos, & Toraldo (2003)
10)PVC routing: Resende & Ribeiro (2003)
11)Phylogenetic trees: Ribeiro & Vianna (2003)
September 2003 GRASP and path-relinking: Advances and applications
39/82 CEMAPRE
GRASP with path-relinking• P is a set (pool) 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 2003 GRASP and path-relinking: Advances and applications
40/82 CEMAPRE
Local access network design
• Design a local access network taking into account tradeoff between:– cost of network– revenue potential of network
September 2003 GRASP and path-relinking: Advances and applications
41/82 CEMAPRE
Local access network design
• Build an optical fyber network to provide large bandwidth services to commercial and residential users, taking into account the construction costs (fyber laying) and the potential revenues obtained by providing the services to the selected clients (loss of revenue in the case of clients which are not served).
• Prize-collecting Steiner tree problem: application developped for AT&T
September 2003 GRASP and path-relinking: Advances and applications
42/82 CEMAPRE
Local access network design residen
ce(revenue)
Street (cost)
September 2003 GRASP and path-relinking: Advances and applications
43/82 CEMAPRE
potential equipmentlocation (cost)
coverage area of equipment
September 2003 GRASP and path-relinking: Advances and applications
44/82 CEMAPRE
area of coverage
potential equipmentlocation
Locate p boxes to maximize revenue of covered residences.
September 2003 GRASP and path-relinking: Advances and applications
45/82 CEMAPRE
equipment box
Assign to each equipment box the total revenueof residences it covers for which there isno closer box. This is the prize.
September 2003 GRASP and path-relinking: Advances and applications
46/82 CEMAPRE
Solve prize-collecting Steiner tree problem
Maximize prize collected minus edge costs:
Here all prizes are collected.
September 2003 GRASP and path-relinking: Advances and applications
47/82 CEMAPRE
Solve prize collecting Steiner tree problem
Maximize prize collected minus edge costs:
Here not all prizes are collected.
September 2003 GRASP and path-relinking: Advances and applications
48/82 CEMAPRE
Prize collecting Steiner tree problem
• Typical dimension: 20,000 to 100,000 nodes.• Use GRASP for maximum covering to locate
equipment boxes.• Compute lower bounds with cutting planes
algorithm of Lucena & Resende (2000).• Compute solutions (upper bounds) with
GRASP (local search with perturbations) algorithm of Canuto, Resende, & Ribeiro (2001).
September 2003 GRASP and path-relinking: Advances and applications
49/82 CEMAPRE
Prize collecting Steiner tree problem• Construction phase (each iteration):
– Memory-based perturbation of original costs to diversify previously obtained solutions.
– Determine nodes to be collected using Goemans & Williamson (1996) 2-opt primal-dual algorithm.
– Connect the selected nodes by a minimum spanning tree.
• Local search: add/drop nodes to/from the current tree.
• PR: nodes in one solution but not in the other
September 2003 GRASP and path-relinking: Advances and applications
50/82 CEMAPRE
Prize collecting Steiner tree problem
• Computational results:– 114 test problems (up to 1,000 nodes and 25,000
edges)– Heuristic found:
•96 out of 104 known optimal solutions.•Solution within 1% of lower bound for 104 of 114 problems.
– PR more than doubled the number of optima found.– Cutting plane algorithm produces tight lower bounds
but running times are very high for the largest instances (days, even weeks), while the GRASP heuristic averaged less than 3 hours for the same instances.
September 2003 GRASP and path-relinking: Advances and applications
51/82 CEMAPRE
Summary• The need for heuristics• Basic GRASP algorithm• Construction phase• Enhanced construction strategies• Local search• Path-relinking• GRASP with path-relinking• Variants of GRASP with path-relinking• Concluding remarks
September 2003 GRASP and path-relinking: Advances and applications
52/82 CEMAPRE
• Probability distribution of time-to-target-solution-value: – Aiex, Resende, & Ribeiro (2002) and Aiex,
Binato, & Resende (2003) have shown experimentally that for both pure GRASP and GRASP with path-relinking the time-to-target-solution-value follows a two-parameter exponential distribution.
– Consequence: linear speedups for independent parallel implementations
Time-to-target-value plots
September 2003 GRASP and path-relinking: Advances and applications
53/82 CEMAPRE
• Probability distribution of time-to-target-solution-value: experimental plots
• 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.
Time-to-target-value plots
September 2003 GRASP and path-relinking: Advances and applications
54/82 CEMAPRE
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
prob
abili
ty
time to target solution value (seconds)
0
1
2
3
4
5
6
0 1 2 3 4 5 6
mea
sure
d tim
es
exponential quantiles
Random variable time-to-target-solution value fits a two-parameter exponential distribution.Therefore, one should expect approximate linear speedup in a straightforward (independent) parallel implementation.
Time-to-target-value plots
September 2003 GRASP and path-relinking: Advances and applications
55/82 CEMAPRE
• 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)
S T
TS
S T
S T
Variants of GRASP + PR
September 2003 GRASP and path-relinking: Advances and applications
56/82 CEMAPRE
• 2-path network design problem:– Graph G=(V,E) with edge weights we and set D
of origin-destination pairs (demands): find a minimum weighted subset of edges E’ E containing a 2-path (path with at most two edges) 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
2-path network design problem
September 2003 GRASP and path-relinking: Advances and applications
57/82 CEMAPRE
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 10 100 1000 10000
prob
abili
ty
time to target value (seconds)
GRASPG+PR(F)G+PR(B)
G+PR(BF)
2-path network design problemEach variant: 200 runs for one instance of 2PNDP
Sun Sparc Ultra 1
80 nodes, 800 pairs,target=588
September 2003 GRASP and path-relinking: Advances and applications
58/82 CEMAPRE
• 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%
2-path network design problem
September 2003 GRASP and path-relinking: Advances and applications
59/82 CEMAPRE
• More recently:– G+PR(M): mixed back and forward
strategyT: elite solution S: local search
– Path-relinking with local search– Randomized path-relinking
TS
Variants of GRASP + PR
September 2003 GRASP and path-relinking: Advances and applications
60/82 CEMAPRE
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 10 100 1000 10000
prob
abili
ty
time to target value (seconds)
GRASPG+PR(F)G+PR(B)
G+PR(BF)G+PR(M)
2-path network design problemEach variant: 200 runs for one instance of 2PNDP
Sun Sparc Ultra 1
80 nodes, 800 pairs,target=588
September 2003 GRASP and path-relinking: Advances and applications
61/82 CEMAPRE
Instance
GRASP
G+PR(F)
G+PR(B)
G+PR(FB)
G+PR(M)
100-3 773 762 756 757 754100-5 756 742 739 737 728200-3 1564 1523 1516 1508 1509200-5 1577 1567 1543 1529 1531300-3 2448 2381 2339 2356 2338300-5 2450 2364 2328 2347 2322400-3 3388 3311 3268 3227 3257400-5 3416 3335 3267 3270 3259500-3 4347 4239 4187 4170 4187500-5 4362 4263 4203 4211 4200
10 runs, same computation time for each variant, best solution found
2-path network design problem
September 2003 GRASP and path-relinking: Advances and applications
62/82 CEMAPRE
• Effectiveness of G+PR(M): – 100 small instances with 70
nodes generated as in Dahl and Johannessen (2000) for comparison purposes.
– Statistical test t for unpaired observations
– GRASP finds better solutions with 40% of confidence (unpaired observations and many optimal solutions):
G+PR(M)
Sample A
D&J Sampl
e B
Size 100 30Mean 443.7
(-2.2%)
453.7
Std. dev.
40.6 61.6
2-path network design problem
Ribeiro & Rosseti (2002)
September 2003 GRASP and path-relinking: Advances and applications
63/82 CEMAPRE
• Effectiveness of path-relinking to improve and speedup the pure GRASP.
• Strategies using the backwards component are systematically better.
2-path network design problem
September 2003 GRASP and path-relinking: Advances and applications
64/82 CEMAPRE
PVC routing• Frame relay service offers virtual private
networks to customers by providing long-term private virtual circuits (PVCs) between customer endpoints on a backbone network.
• Routing is done either automatically by switch or by the network designer without any knowledge of future requests.
• Over time, these decisions cause inefficiencies in the network and occasionally offline rerouting (grooming) of the PVCs is needed: – integer multicommodity network flow problem:
Resende & Ribeiro (2003)
September 2003 GRASP and path-relinking: Advances and applications
65/82 CEMAPRE
PVC routing
September 2003 GRASP and path-relinking: Advances and applications
66/82 CEMAPRE
PVC routing
September 2003 GRASP and path-relinking: Advances and applications
67/82 CEMAPRE
PVC routing
September 2003 GRASP and path-relinking: Advances and applications
68/82 CEMAPRE
PVC routing
September 2003 GRASP and path-relinking: Advances and applications
69/82 CEMAPRE
PVC routingmax capacity = 3
September 2003 GRASP and path-relinking: Advances and applications
70/82 CEMAPRE
PVC routingmax capacity = 3very long path!
September 2003 GRASP and path-relinking: Advances and applications
71/82 CEMAPRE
PVC routingmax capacity = 3very long path!
reroute
September 2003 GRASP and path-relinking: Advances and applications
72/82 CEMAPRE
PVC routingmax capacity = 3
September 2003 GRASP and path-relinking: Advances and applications
73/82 CEMAPRE
PVC routingmax capacity = 3feasible and
optimal!
September 2003 GRASP and path-relinking: Advances and applications
74/82 CEMAPRE
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
10 100 1000 10000 100000 1e+06
GGPRfGPRb
GPRfb
time (seconds)
Prob
abilit
yEach variant: 200 runs for one instance of PVC routing problem
(60 nodes, 498 edges, 750 origin-destination pairs)
PVC routing
SGI Challenge 196 MHz
September 2003 GRASP and path-relinking: Advances and applications
75/82 CEMAPRE
PVC routing10 runs 10 seconds 100 secondsVariant best avera
ge best average
GRASP 126603 126695 126228 126558
G+PR(F) 126301 126578 126083 126229
G+PR(B) 125960 126281 125666 125883
G+PR(BF) 125961 126307 125646 125850
September 2003 GRASP and path-relinking: Advances and applications
76/82 CEMAPRE
PVC routing10 runs 10 seconds 100 secondsVariant best avera
ge best average
GRASP 126603 126695 126228 126558
G+PR(F) 126301 126578 126083 126229
G+PR(B) 125960 126281 125666 125883
G+PR(BF) 125961 126307 125646 125850
September 2003 GRASP and path-relinking: Advances and applications
77/82 CEMAPRE
PVC routingGRASP + PR backwards: four increasingly difficult target values
Same behavior, plots drift to the right for more difficult targets
SGI Challenge 196 MHz
September 2003 GRASP and path-relinking: Advances and applications
78/82 CEMAPRE
• Post-optimization “evolutionary” strategy:a) Start with pool P0 found at end of GRASP and set k =
0.b) Combine with path-relinking all pairs of solutions in Pk.c) Solutions obtained by combining solutions in Pk are
added to a new pool Pk+1 following same constraints for updates as before.
d) If best solution of Pk+1 is better than best solution of Pk, then set k = k + 1, and go back to step (b).
• Succesfully used by Ribeiro, Uchoa, & Werneck (2002) (Steiner) and Resende & Werneck (2002) (p-median)
GRASP with path-relinking
September 2003 GRASP and path-relinking: Advances and applications
79/82 CEMAPRE
Summary• The need for heuristics• Basic GRASP algorithm• Construction phase• Enhanced construction strategies• Local search• Path-relinking• GRASP with path-relinking• Variants of GRASP with path-relinking• Concluding remarks
September 2003 GRASP and path-relinking: Advances and applications
80/82 CEMAPRE
Concluding remarks (1/2)• Path-relinking adds memory and
intensification mechanisms to GRASP, systematically contributing to improve solution quality: – better solutions in smaller times– some implementation strategies appear to
be more effective than others). – mixed path-relinking strategy is very
promising– backward relinking is usually more effective
than forward– bidirectional relinking does not necessarily
pays the additional computation time
September 2003 GRASP and path-relinking: Advances and applications
81/82 CEMAPRE
Concluding remarks (2/2)• Difficulties:
– How to deal with infeasibilities along the relinking procedure?
– How to apply path-relinking in “partitioning” problems such as graph-coloring, bin packing, and others?
• Other applications of path-relinking:– VNS+PR: Festa, Pardalos, Resende, & Ribeiro
(2002)– PR as a generalized optimized crossover in
genetic algorithms: Ribeiro & Vianna (2003) • Effective parallel strategies based on path-
relinking to introduce memory and processor cooperation (robustness and linear speedups).
September 2003 GRASP and path-relinking: Advances and applications
82/82 CEMAPRE
Slides, publications, and acknowledgements
• Slides of this talk can be downloaded from: http://www.inf.puc-rio/~celso/talks
• Papers about GRASP, path-relinking, and their applications available at:http://www.inf.puc-rio.br/~celso/publicacoes
• Joint work done mostly with Maurício Resende (AT&T Labs) and M.Sc. and Ph.D. students from PUC-Rio, who are all gratefully acknowledged: S. Canuto, M. Souza, M. Prais, S. Martins, D. Vianna, R. Aiex, R. Werneck, E. Uchoa, and I. Rosseti.