Solving Strategies using a Hybridization Model for Local Search and Constraint Propagation

1. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Solving Strategies using a Hybridization Model
for Local Search and Constraint Propagation
ACM SAC 2005, Santa Fe, New Mexico, USA
Tony Lambert and Éric Monfroy and Frédéric Saubion
LERIA, Université de Angers, France
and
LINA, Université de Nantes, France
March 16, 2005
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2. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Outline
• Hybridization
• Constraint propagation
• Local search
• Hybrid model
• Some results
• Conclusion
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3. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Hybridization for CSP
• complete methods (such as propagation + split)
• complete exploration of the search space
• detects if no solution
• generally slow for hard combinatorial problems
• global optimum
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4. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Hybridization for CSP
• complete methods (such as propagation + split)
• complete exploration of the search space
• detects if no solution
• generally slow for hard combinatorial problems
• global optimum
• incomplete methods (such as local search)
• focus on some “promising” parts of the search space
• does not answer to unsat. problems
• no guaranteed global optimum
• “fast” to find a “good” solution
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5. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Hybridization : getting the best of both methods
• But, generally :
• Ad-hoc systems
• Master-slaves approaches
• Coarse grain cooperation
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6. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Hybridization : getting the best of both methods
• But, generally :
• Ad-hoc systems
• Master-slaves approaches
• Coarse grain cooperation
• Idea :
• Fine grain control
• More strategies
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7. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Hybridization : getting the best of both methods
• But, generally :
• Ad-hoc systems
• Master-slaves approaches
• Coarse grain cooperation
• Idea :
• Fine grain control
• More strategies
• Technique :
• Decomposing solvers into basic functions
• Adapting chaotic iterations for hybrid solving
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8. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
CSP (Constraint Satisfaction Problem)
C2
Y
C1 U
Variables
X
C4 C5
Constraints
Z T
C3
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9. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Problems are modeled as CSP (X , D, C)
Variable (X )
Set of variable domains (D)
Dx = {a; b; c; . . .}
Dy = {a; b; c; . . .}
Dz = {a; b; c; . . .}
...
Set of constraints (C)
C1 : X ≤ Y ∗ 3
C2 : Z = X − Y
...
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10. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Solving CSP with a complete method
Constraint
Propagation
Enumeration
Splitting
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11. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Constraint Propagation
Arc consistency : C(X , Y )
a a
b b Values
c c
D D
X Y
Consistent
pairs of values
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12. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Constraint Propagation
Arc consistency : C(X , Y )
Removed by arc
consistency
a a
b b Values
c c
D D
X Y
Consistent
pairs of values
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13. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Constraint Propagation
Propagation
Local consistency
Propagation
inconsistency
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14. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Constraint propagation and domain splitting
Propagation
Splitting
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15. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Constraint propagation and domain splitting
Propagation
Splitting
Propagation
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16. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Abstract Model K.R. Apt [CP99]
Abstraction
Reduction functions
Constraint
Application
of functions
CSP
Fixed point
Partial Ordering
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17. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
A Generic Algorithm
F = { set of split and propagation functions}
X = initial CSP
G=F X=
While G = ∅
choose g ∈ G
G = G − {g}
G = G ∪ update(G, g, X )
X = g(X )
EndWhile
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18. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Our Purpose : Hybrid Model
• Integration of local search
• Use of an existing theoretical model for CSP solving
• Definition of the solving process
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19. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Local search
• Search space : set of possible configurations
• Tools : neighborhood and evaluation function
Set of
possible
configurations
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20. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Local search
• Search space : set of possible configurations
• Tools : neighborhood and evaluation function
Neighborhood
of s Set of
possible
configurations
s3
s4
s6
s1 s2
s5
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21. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Local search
• Search space : set of possible configurations
• Tools : neighborhood and evaluation function
Neighborhood
of s Set of
possible
configurations
s3
s4
s6
s1 s2
s5
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22. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Local search
• Search space : set of possible configurations
• Tools : neighborhood and evaluation function
Neighborhood
of s Set of
possible
configurations
LS(p) = p’
p=( s 1 , s 2 , ..., sn)
s3 p’ = ( s 1 , s 2 , ...,
s4 s n , s n+1 )
s6
s1 s2
s5
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23. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Integrating CSP Resolution Techniques in a Hybrid Model
• Extend the existing Apt s theoretical model for CSP
solving
• Fixpoint computation on a partial ordering
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24. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
The theoretical model for CSP solving
Partial ordering :
Ω1
Application of a reduction :
Ω2
domain reduction function or
splitting function
Ω3
Set of CSP
Ω4
Ω5
Terminates with a fixed point : set of solutions or inconsistent CSP
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25. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
The theoretical model for CSP solving
Reduction : by constraint propagation
Ω Ω’
Ω CSP1 CSP2 CSP3
Constraint Propagation
Ω CSP1 CSP2’ CSP3
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26. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
The theoretical model for CSP solving
Reduction : by domain splitting
Ω Ω’
Ω CSP1 CSP2 CSP3
Split
Ω CSP1 CSP2’ CSP2’’ CSP2’’’ CSP3
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27. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
LS process as moves on partial ordering
Moves on a partial ordering
p
1
Function to pass from one LS
state to another
p
2
State of LS
p
3
p
4
p
5
Terminates with a fixed point : local minimum, maximum number of iteration
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28. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
LS Ordering
Characteristics of a LS path
• Notion of Solution
• Maximum Length
Operational (computational) point of view : path = samples
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29. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Integration of samples for local search into the CSP
• A sample :
• Depends on a CSP
• Corresponds to a LS state
• An SCSP contains :
• A set of domains ;
• A set of constraints ;
• A (list of) sample for local search.
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30. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Hybridization model
Ordering over SCSP
Ω1
Application of a reduction :
Ω2
function :
− domain reduction,
− splitting,
Ω3
− or LS step.
Set of SCSP
Ω4
Ω5
A solution before the end of the process, given by LS
Terminates with a fixed point : set of solutions or inconsistent SCSP
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31. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Hybridization model
Ordering over SCSP
Ψ = C; D; p
with :
Ψ Ψ
Ψ = C; D ; p
D⊆D
if :
D = D and p p
or if :
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32. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Algorithm
Same Generic Algorithm
X = initial SCSP
G=F
While G = ∅
choose g ∈ G
G = G − {g}
G = G ∪ update(G, g, X )
X = g(X )
EndWhile
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33. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
General Process
Application
of functions
Sets of
Constraint
SCSP
LS solution
Global
Fixed point
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34. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Algorithm
• In the chaotic iteration framework
• Finite domains (sets of SCSP)
• Monotonic functions (LS move, domain reduction, splitting)
• Decreasing functions
• → termination and computation of fixed point (solution of
CP)
• Practically : can stop before with a LS solution
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35. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Examples
• SEND + MORE = MONEY
• Zebra
• Golomb
• Magic square
• Langford Number
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36. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Modeling Strategies through Reduction Functions
• Reduction / Split / LS functions
• Choose a / several SCPS to apply functions
• Choose a / several domains (Prop. - Split)
• Tests : Random - Depth first - FC
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37. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Strategy : ratio LS-CP
• Ratio of LS functions applied and CP functions applied :
Triple(red%,split%,ls%)
• 10% of split compared to reduction
• CP alone : (90,10,0)
• LS alone : (0,0,100)
• LS strategy : tabu
• Choose : LS, reduction, or split but keep the given ratios
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38. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Langford number 4 2
800
700
600
Number of operations
500
400
300
200 Average
100
standard deviation
0
10 20 30 40 50 60 70 80 90 100
Propagation percentage
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39. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
SEND + MORE = MONEY
2000
1800
1600
1400
Number of operations
1200
standard deviation
1000
800
600
Average
400
200
10 20 30 40 50 60 70 80 90 100
Propagation percentage
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40. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Problem S+M LN42 Zebra M. square Golomb
Rate FC 70-80 15 - 25 60-70 30-45 30 - 40
TAB.: Best range of propagation rate (α) to compute a solution
Strategy Method S+M LN42 M. square Golomb
Random LS 1638 383 3328 3442
CP+LS 1408 113 892 909
CP 3006 1680 1031 2170
D-First LS 1535 401 3145 3265
CP+LS 396 95 814 815
CP 1515 746 936 1920
FC LS 1635 393 3240 3585
CP+LS 22 192 570 622
CP 530 425 736 1126
TAB.: Avg. nb. of operations to compute a first solution
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41. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Conclusion
• A generic model for hybridizing complete (CP) and
incomplete (LS) methods
• Implementation of modules working on the same structure
• Complementarity of methods
• Design of strategies
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42. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION
Future work
• Adaptation of the model for optimization
• Study of other methods (G.A.)
• Design of strategies using composition operators
• A generic implementation
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