Hybridization of complete and incomplete methods to solve CSP

1. Hybridization of
complete and
incomplete
methods to solve
CSP
Hybridization of complete and
Tony Lambert
incomplete methods to solve CSP
Solving CSP
Hybrid solving :
need a framework
Integrate split in GI
Tony Lambert
Theoretical model
for hybridization
CP + LS
LERIA, Angers, France
Theoretical model
for hybridization LINA, Nantes, France
CP + GA
To an uniform
October 27th, 2006
framework CP +
GA + LS
Conclusion and
future works
Rapporteurs : François FAGES Directeur de Recherche, INRIA Rocquencourt
Bertrand NEVEU, Ingénieur en Chef des Ponts et Chaussées, HDR
Examinateurs : Steven PRESTWICH, Associate Professor, University College Cork
Jin-Kao HAO, Professeur à l’Université d’Angers
Directeurs de thèse : Éric MONFROY, Professeur à l’Université de Nantes
Frédéric SAUBION, Professeur à l’Université d’Angers
1 / 86

2. Hybridization of
Constraint programming process
complete and
incomplete
methods to solve
CSP
Formulate the problem with constraints as a CSP
Tony Lambert
constraint : a relation on variables and their domains
Solving CSP
Constraint Satisfaction Problem (CSP) : a set of
Hybrid solving :
need a framework
constraints together with a set of variable domains
Integrate split in GI
Theoretical model
CSP model
for hybridization
CP + LS
Theoretical model
C2
Y
for hybridization
X = {x1 , . . . , xn } set of n
C1
CP + GA U
To an uniform
variables,
X
C4 C5
framework CP +
GA + LS
D = {Dx1 , . . . , Dxn } set
Conclusion and
Z T
of n domains,
C3
future works
C = {c1 , . . . , cm } set of
m constraints.
Variables Constraints
2 / 86

3. Hybridization of
Problems are modeled as CSPs (X , D, C)
complete and
incomplete
methods to solve
CSP
Tony Lambert
Some variables to represent objects
Solving CSP
(X = {X1 , . . . , Xn })
Hybrid solving :
need a framework
Integrate split in GI
Domains over which variables can range
Theoretical model
for hybridization
(D = D1 × . . . × Dn )
CP + LS
Theoretical model
for hybridization
Some constraints to set relation between objects
CP + GA
To an uniform
framework CP +
C1 : X ≤ Y ∗ 3
GA + LS
C2 : Z = X − Y
Conclusion and
future works
...
3 / 86

4. Hybridization of
Example : Sudoku problem
complete and
incomplete
methods to solve
6 9 2 1 4
CSP
3 6 9 8
Tony Lambert
8 2
Solving CSP
1 9 7 2
Hybrid solving :
need a framework 9 5
Integrate split in GI 8 4 6 7
Theoretical model 9 5
for hybridization
CP + LS 2 3 5 1
Theoretical model 1 5 6 3 7
for hybridization
CP + GA
1 2 3 4 5 6 7 8 9
To an uniform
framework CP +
GA + LS
X = { all the cells in the grid},
Conclusion and
future works
D = { to each cell a value in [1..9]},
C = { set of alldiff constraints : all digits appears only
once in each row ; once in each column and once in
each 3 × 3 square the grid has been subdivided}
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5. Hybridization of
Example : 8 Queens
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Hybrid solving :
need a framework
Integrate split in GI
Theoretical model
for hybridization
CP + LS
Theoretical model
for hybridization
CP + GA
To an uniform
framework CP +
GA + LS
X = {x1 , ...x8 } Columns,
Conclusion and
future works
D = {Dx1 , ..., Dx8 } for each columns, the row the
queen is placed Dxi = [1..8],
C = { not 2 queens in the same row or same
diagonal}
5 / 86

6. Hybridization of
CSP solving
complete and
incomplete
methods to solve
A solution
CSP
Tony Lambert
Given a search space S = Dx1 × ... × Dxn
an assignment s is a solution if :
Solving CSP
Hybrid solving :
s ∈ S and
need a framework
∀c ∈ C, s ∈ c
Integrate split in GI
Theoretical model
for hybridization
Solving a CSP can be :
CP + LS
compute whether the CSP has a solution
Theoretical model
for hybridization
(satisfiability)
CP + GA
To an uniform
framework CP +
find A solution
GA + LS
Conclusion and
find ALL solutions
future works
find optimal solutions (global optimum)
find A good solution (local optimum)
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7. Hybridization of
Outline
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
1
Solving CSP
Hybrid solving : need a framework
2
Hybrid solving :
need a framework
Integrate split in GI
Integrate split in GI
3
Theoretical model
for hybridization
CP + LS
Theoretical model for hybridization CP + LS
4
Theoretical model
for hybridization
CP + GA
Theoretical model for hybridization CP + GA
5
To an uniform
framework CP +
GA + LS
To an uniform framework CP + GA + LS
6
Conclusion and
future works
Conclusion and future works
7
7 / 86

8. Hybridization of
complete and
incomplete
methods to solve
Solving CSP
1
CSP
Tony Lambert
Hybrid solving : need a framework
2
Solving CSP
Propagation + Split
Local Search
Genetic Algorithms
Integrate split in GI
3
Hybrid solving :
need a framework
Integrate split in GI
Theoretical model for hybridization CP + LS
4
Theoretical model
for hybridization
CP + LS
Theoretical model for hybridization CP + GA
5
Theoretical model
for hybridization
CP + GA
To an uniform framework CP + GA + LS
6
To an uniform
framework CP +
GA + LS
Conclusion and
Conclusion and future works
7
future works
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9. Hybridization of
complete/incomplete methods
complete and
incomplete
methods to solve
CSP
Tony Lambert
complete methods (such as propagation + split)
complete exploration of the search space
Solving CSP
Propagation + Split
Local Search
detect if no solution
Genetic Algorithms
Hybrid solving :
global optimum
need a framework
Integrate split in GI
generally slow for hard combinatorial problems
Theoretical model
for hybridization
CP + LS
incomplete methods (such as local search)
Theoretical model
for hybridization
focus on some “promising” parts of the search space
CP + GA
To an uniform
do not detect if no solution
framework CP +
GA + LS
no global optimum
Conclusion and
future works
“fast” to find a “good” solution
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10. Hybridization of
Solving CSP
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Propagation + Split
Local Search
Genetic Algorithms
Hybrid solving :
Methods
need a framework
Integrate split in GI
Complete methods
Theoretical model
Incomplete methods
for hybridization
CP + LS
Theoretical model
for hybridization
CP + GA
To an uniform
framework CP +
GA + LS
Conclusion and
future works
10 / 86

11. Hybridization of
Complete methods
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Propagation + Split
Local Search
General methods that can be adapted to several
Genetic Algorithms
Hybrid solving :
types of constraints and several types of variables
need a framework
search methods : to explore the search space
Integrate split in GI
Theoretical model
constraint propagation algorithms : to repeatedly
for hybridization
CP + LS
remove inconsistent values from domains of
Theoretical model
variables
for hybridization
CP + GA
To an uniform
framework CP +
GA + LS
Conclusion and
future works
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12. Hybridization of
First approach
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Complete Methods
Propagation + Split
Local Search
Genetic Algorithms
Search space : Dx1 × · · · × Dxn
Hybrid solving :
Enumerate all assignments
need a framework
Integrate split in GI
Theoretical model
Backtracking
for hybridization
CP + LS
search (backtrack)
Theoretical model
for hybridization
Select variables
CP + GA
Split / enumeration
To an uniform
framework CP +
GA + LS
Conclusion and
future works
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13. Hybridization of
Backtracking for 4 - Queens
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Propagation + Split
Local Search
Genetic Algorithms
Hybrid solving :
need a framework
Integrate split in GI
Theoretical model
for hybridization
CP + LS
Theoretical model
for hybridization
CP + GA
To an uniform
framework CP +
GA + LS
Conclusion and
future works
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14. Hybridization of
Constraint propagation : reducing domains
complete and
incomplete
methods to solve
CSP
Tony Lambert
Generally :
Solving CSP
Propagation + Split
reduce domains using constraint and domains
Local Search
Genetic Algorithms
→ reduce the search space
Hybrid solving :
need a framework
Integrate split in GI
Generic domain reduction :
Theoretical model
for hybridization
given a constraint C over x1 , . . . , xn with domains
CP + LS
D1 , . . . , Dn
Theoretical model
for hybridization
select a variable xi reduce its domain
CP + GA
To an uniform
delete from Di all values for xi that do not participate
framework CP +
GA + LS
in a solution of C
Conclusion and
future works
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15. Hybridization of
Forward Checking
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Propagation + Split
Local Search
Genetic Algorithms
Hybrid solving :
need a framework
Integrate split in GI
Theoretical model
for hybridization
CP + LS
Theoretical model
for hybridization
CP + GA
To an uniform
framework CP +
GA + LS
Conclusion and
future works
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16. Hybridization of
Constraint propagation
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Propagation + Split
Local Search
Genetic Algorithms
constraint propagation mechanism : repeatedly
Hybrid solving :
need a framework
reduce domains
Integrate split in GI
replace a CSP by a CSP which is :
Theoretical model
for hybridization
equivalent (same set of solutions)
CP + LS
“smaller” (domains are reduced)
Theoretical model
for hybridization
CP + GA
To an uniform
framework CP +
GA + LS
Conclusion and
future works
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17. Hybridization of
Constraint propagation framework
complete and
incomplete
methods to solve
CSP
Tony Lambert
Can be seen as a fixed point of application of
Solving CSP
reduction functions
Propagation + Split
Local Search
reduction function to reduce domains or constraints
Genetic Algorithms
Hybrid solving :
can be seen as an abstraction of the constraints by
need a framework
Integrate split in GI
reduction functions
Theoretical model
for hybridization
Chaotic iteration
CP + LS
Theoretical model
Compute a limit of a set of functions [Cousot and
for hybridization
CP + GA
Cousot 77]
To an uniform
monotonic and inflationary functions in a generic
framework CP +
GA + LS
algorithm to achieve consistency [Apt 97]
Conclusion and
future works
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18. Hybridization of
Abstract model K.R. Apt [CP99]
complete and
incomplete
For propagation (based on chaotic iterations)
methods to solve
CSP
Tony Lambert
Abstraction
Solving CSP
Reduction functions
Constraint
Propagation + Split
Local Search
Genetic Algorithms
Hybrid solving :
need a framework
Integrate split in GI
Theoretical model
for hybridization
CP + LS
Theoretical model
Application
for hybridization
CP + GA
of functions
CSP
To an uniform
Fixed point
framework CP +
GA + LS
Partial Ordering
Conclusion and
future works
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19. Hybridization of
Partial ordering and functions
complete and
incomplete
methods to solve
Partial Ordering
CSP
Tony Lambert
Given a CSP (X , D, C)
P(D) : all possible subset from D
Solving CSP
Propagation + Split
Local Search
: subset relation ⊇
Genetic Algorithms
Hybrid solving :
need a framework
=⇒ (P(D), ) is a partial ordering
Integrate split in GI
Theoretical model
Functions
for hybridization
CP + LS
Given a set F of functions on D, every f ∈ F is :
Theoretical model
for hybridization
inflationary : x f (x)
CP + GA
To an uniform
monotonic : x y implies f (x) f (y )
framework CP +
GA + LS
idempotent : f (f (x)) = f (x)
Conclusion and
future works
=⇒ Every sequence of elements d0 d1 ... with
dj = fij (dj−1 ) stabilizes to a fix point.
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20. Hybridization of
Example of reduction functions (1)
complete and
incomplete
methods to solve
Constraint x < y
CSP
5
Tony Lambert
3
6
4
Solving CSP
7
Propagation + Split
5
Local Search
8
Genetic Algorithms
6
9
Hybrid solving :
7
need a framework
10
Integrate split in GI
D D
x y
Theoretical model
for hybridization
CP + LS
Arc_consistence on Dx and Arc_consistence on Dy
Theoretical model
5 5
for hybridization
CP + GA
3 3
6 6
To an uniform
4 4
7 7
framework CP +
GA + LS 5 5
8 8
Conclusion and 6 6
9 9
future works
7 7
10 10
D D D D
x y x y
20 / 86

21. Hybridization of
Example of reduction functions (2)
complete and
incomplete
methods to solve
CSP
Arc consistency
Tony Lambert
Consider a constraint x < y with Dx = [5..10] and
Solving CSP
Dy = [3..7] then 2 functions D → D :
Propagation + Split
Local Search
Arc_consistence 1 :
Genetic Algorithms
Hybrid solving :
need a framework
(Dx , Dy ) → (Dx , Dy )
Integrate split in GI
Theoretical model
s.t.
for hybridization
CP + LS
Dx = {a ∈ Dx | ∃b ∈ Dy , a < b}
Theoretical model
for hybridization
Arc_consistence 2 :
CP + GA
To an uniform
framework CP +
(Dx , Dy ) → (Dx , Dy )
GA + LS
Conclusion and
future works
s.t.
Dy = {b ∈ Dy | ∃a ∈ Dx , a < b}
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22. Hybridization of
A Generic Algorithm to reach fixpoint
complete and
incomplete
methods to solve
CSP
Tony Lambert
for constraint propagation : ordering on size of domains
Solving CSP
Propagation + Split
work on a CSP
Local Search
F = { set of propagation functions}
Genetic Algorithms
Hybrid solving :
X = initial CSP
need a framework
G=F
Integrate split in GI
While G = ∅
Theoretical model
for hybridization
choose g ∈ G
CP + LS
G = G − {g}
Theoretical model
for hybridization
G = G ∪ update(G, g, X )
CP + GA
To an uniform
X = g(X )
framework CP +
EndWhile
GA + LS
Conclusion and
future works
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23. Hybridization of
Solving CSP
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Propagation + Split
Local Search
Genetic Algorithms
Hybrid solving :
Methods
need a framework
Integrate split in GI
Complete methods
Theoretical model
Incomplete methods
for hybridization
CP + LS
Theoretical model
for hybridization
CP + GA
To an uniform
framework CP +
GA + LS
Conclusion and
future works
23 / 86

24. Hybridization of
Second Approach
complete and
incomplete
methods to solve
CSP
Tony Lambert
Incomplete methods
Solving CSP
heuristics algorithms
Propagation + Split
Local Search
Metaheuristics
Genetic Algorithms
Hybrid solving :
need a framework
two families
Integrate split in GI
Local search
Theoretical model
for hybridization
Simulated annealing [Kirkpatrick et al, 1983]
CP + LS
Tabu Search [Glover, 1986]
Theoretical model
for hybridization ...
CP + GA
Evolutionary Algorithms
To an uniform
framework CP +
Genetic Algorithms [Holland, 1975]
GA + LS
Genetic programming [Koza, 1992]
Conclusion and
future works
...
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25. Hybridization of
Incomplete methods
complete and
incomplete
methods to solve
CSP
Tony Lambert
Definitions
Explore a D1 × D2 × · · · × Dn search space
Solving CSP
Propagation + Split
Local Search
Move from neighbor to neighbor (resp. generation to
Genetic Algorithms
generation) thanks to an evaluation function
Hybrid solving :
need a framework
Intensification
Integrate split in GI
Diversification
Theoretical model
for hybridization
CP + LS
Properties :
Theoretical model
for hybridization
focus on some “promising” parts of the search space
CP + GA
does not answer to unsat. problems
To an uniform
framework CP +
GA + LS
no guaranteed
Conclusion and
“fast” to find a “good” solution
future works
25 / 86

26. Hybridization of
Local search (1)
complete and
incomplete
methods to solve
Search space : set of possible configurations
CSP
Tony Lambert
Tools : neighborhood and evaluation function
Solving CSP
Propagation + Split
Local Search
Genetic Algorithms
Hybrid solving :
need a framework
Set of
Integrate split in GI
possible
configurations
Theoretical model
for hybridization
CP + LS
Theoretical model s 3 s
for hybridization 4
CP + GA s 6
s s
1 2 s 5
To an uniform
framework CP +
GA + LS
Conclusion and
future works
26 / 86

27. Hybridization of
Local search (2)
complete and
incomplete
methods to solve
Search space : set of possible configurations
CSP
Tony Lambert
Tools : neighborhood and evaluation function
Solving CSP
Propagation + Split
Local Search
Genetic Algorithms
Hybrid solving :
Neighborhood
need a framework
of s Set of
Integrate split in GI
possible
configurations
Theoretical model
for hybridization
CP + LS
Theoretical model s 3 s
for hybridization 4
CP + GA s 6
s s
1 2 s 5
To an uniform
framework CP +
GA + LS
Conclusion and
future works
27 / 86

28. Hybridization of
Local search (3)
complete and
incomplete
methods to solve
Search space : set of possible configurations
CSP
Tony Lambert
Tools : neighborhood and evaluation function
Solving CSP
Propagation + Split
Local Search
Genetic Algorithms
Hybrid solving :
Neighborhood
need a framework
of s Set of
Integrate split in GI
possible
configurations
Theoretical model
for hybridization
CP + LS
Theoretical model s 3 s
for hybridization 4
CP + GA s 6
s s
1 2 s 5
To an uniform
framework CP +
GA + LS
Conclusion and
future works
28 / 86

29. Hybridization of
Local search (4)
complete and
incomplete
methods to solve
CSP
Search space : set of possible configurations
Tony Lambert
Tools : neighborhood and evaluation function
Solving CSP
Propagation + Split
Local Search
Genetic Algorithms
Hybrid solving :
need a framework Neighborhood
Set of
of s
Integrate split in GI possible
configurations
Theoretical model
for hybridization LS(p) = p’
CP + LS
p=( s ,s , ..., s )
1 2 n
Theoretical model s p’ = ( s ,s , ..., s ,s )
3 1 2
s n n+1
for hybridization 4
CP + GA s 6
s s
1 2
s 5
To an uniform
framework CP +
GA + LS
Conclusion and
future works
29 / 86

30. Hybridization of
Genetic algorithms (1)
complete and
incomplete
methods to solve
CSP
Search space : set of possible configurations
Tony Lambert
Tools : population, crossing, mutations, and
Solving CSP
evaluation function
Propagation + Split
Local Search
Genetic Algorithms
Hybrid solving :
need a framework
Population k
Integrate split in GI
Theoretical model
for hybridization x
1 4 9 4 2 5
CP + LS y
Theoretical model
for hybridization 1 4 2 4 2 9 Population k+1
CP + GA
To an uniform
8 6 2 41 9
framework CP +
GA + LS
Conclusion and
future works
30 / 86

31. Hybridization of
Genetic algorithms (2)
complete and
incomplete
methods to solve
CSP
Search space : set of possible configurations
Tony Lambert
Tools : population, crossing, mutations, and
Solving CSP
evaluation function
Propagation + Split
Local Search
Genetic Algorithms
Hybrid solving :
need a framework
Population k
Integrate split in GI
Theoretical model
for hybridization x
1 4 9 4 2 5
CP + LS y
Theoretical model
for hybridization 1 4 9 4 8 5 Population k+1
CP + GA
To an uniform
framework CP +
GA + LS
Conclusion and
future works
31 / 86

32. Hybridization of
Genetic algorithms (3)
complete and
incomplete
methods to solve
CSP
Search space : set of possible configurations
Tony Lambert
Tools : population, crossing, mutations, and
Solving CSP
evaluation function
Propagation + Split
Local Search
Genetic Algorithms
Hybrid solving :
need a framework
Population k
Integrate split in GI
Croisement
Theoretical model
for hybridization
CP + LS
Mutation
Theoretical model
for hybridization
Population k+1
CP + GA
To an uniform
framework CP +
Nouvelle Population
GA + LS
Conclusion and
future works
32 / 86

33. Hybridization of
complete and
incomplete
methods to solve
Solving CSP
1
CSP
Tony Lambert
Hybrid solving : need a framework
2
Solving CSP
Hybrid solving :
need a framework
Integrate split in GI
3
Why hybridization
complete/incomplete
Hybridization : getting the
best of the both
Theoretical model for hybridization CP + LS
Integrate split in GI 4
Theoretical model
for hybridization
CP + LS
Theoretical model for hybridization CP + GA
5
Theoretical model
for hybridization
CP + GA
To an uniform framework CP + GA + LS
6
To an uniform
framework CP +
GA + LS
Conclusion and future works
7
Conclusion and
future works
33 / 86

34. Hybridization of
Hybridization : getting the best of the both
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Hybrid solving :
need a framework
Efficiency : faster complete solver
Why hybridization
complete/incomplete
Quality : better solutions (better optimum)
Hybridization : getting the
best of the both
Generally :
Integrate split in GI
Theoretical model
Ad-hoc systems (designed from scratch)
for hybridization
CP + LS
Dedicated to a class of problems
Theoretical model
Master-slave approaches (LS for CP, CP for LS)
for hybridization
CP + GA
To an uniform
framework CP +
GA + LS
Conclusion and
future works
34 / 86

35. Hybridization of
Hybridization : Overview
complete and
incomplete
methods to solve
CSP
Tony Lambert
complete and incomplete methods combinations
Solving CSP
Hybrid solving :
need a framework
Why hybridization
complete/incomplete
collaborative combinations
Hybridization : getting the
best of the both
Integrate split in GI
Theoretical model
for hybridization
integrative combinaisons
CP + LS
Theoretical model
for hybridization
CP + GA
Incorporate a complete method in metaheuristics
To an uniform
framework CP +
GA + LS
Conclusion and
Incorporate a metaheuristic in complete methods
future works
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36. Hybridization of
CP for LS
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Neighborhood
Hybrid solving : of s Set of
need a framework
possible
Why hybridization
configurations
complete/incomplete
Hybridization : getting the
best of the both
Integrate split in GI
s
Theoretical model 3 s 4
for hybridization
s
CP + LS 6
s s
1 2 s 5
Reduced
Theoretical model
for hybridization search space
CP + GA
To an uniform
framework CP +
Valid neighborhood
GA + LS
of s
Conclusion and
future works
36 / 86

37. Hybridization of
CP for GA (1)
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Hybrid solving : Set of
need a framework possible
configurations
Why hybridization
complete/incomplete
Hybridization : getting the
best of the both
Integrate split in GI
Theoretical model
for hybridization
CP + LS
Theoretical model
Population
for hybridization
CP + GA
To an uniform
framework CP +
GA + LS
Conclusion and
future works
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38. Hybridization of
CP for GA (2)
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Hybrid solving : Set of
need a framework possible
configurations
Why hybridization
complete/incomplete
Hybridization : getting the
best of the both
Integrate split in GI
Theoretical model
for hybridization
CP + LS
Theoretical model
for hybridization
Feasible population
CP + GA
To an uniform
framework CP +
GA + LS
Conclusion and
future works
38 / 86

39. Hybridization of
Hybridization : getting the best of the both
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Hybrid solving :
need a framework
Idea :
Why hybridization
complete/incomplete
Hybridization : getting the
fine grain hybridization
best of the both
finer strategies
Integrate split in GI
every technique at the same level
Theoretical model
for hybridization
one algorithm squeleton
CP + LS
easier to modify, extend, compare, ...
Theoretical model
for hybridization
CP + GA
To an uniform
framework CP +
GA + LS
Conclusion and
future works
39 / 86

40. Hybridization of
complete and
incomplete
methods to solve
Solving CSP
1
CSP
Tony Lambert
Hybrid solving : need a framework
2
Solving CSP
Hybrid solving :
need a framework
Integrate split in GI
3
Integrate split in GI
Constraint propagation
Domain splitting
Theoretical model for hybridization CP + LS
4
Theoretical model
for hybridization
CP + LS
Theoretical model
Theoretical model for hybridization CP + GA
5
for hybridization
CP + GA
To an uniform
To an uniform framework CP + GA + LS
6
framework CP +
GA + LS
Conclusion and
future works
Conclusion and future works
7
40 / 86

41. Hybridization of
Our purpose :
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Hybrid solving :
Use of an existing theoretical model for CSP solving
need a framework
Definition of the solving process
Integrate split in GI
Constraint propagation
Domain splitting
Theoretical model
Integrate :
for hybridization
CP + LS
Split
Theoretical model
for hybridization
LS
CP + GA
GA
To an uniform
framework CP +
GA + LS
Conclusion and
future works
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42. Hybridization of
Search Tree
complete and
incomplete
methods to solve
CSP
Tony Lambert
CSP1
propagation
Solving CSP
CSP1’
Hybrid solving :
split
need a framework
Integrate split in GI CSP2 CSP3
propagation
Constraint propagation
Domain splitting
CSP2’ CSP3
Theoretical model
split
for hybridization
CP + LS
CSP5
CSP4 CSP3
propagation
Theoretical model
for hybridization
CP + GA CSP5
CSP4 CSP3’
split
To an uniform
framework CP +
CSP6
CSP4 CSP5 CSP7 CSP8
GA + LS
propagation
Conclusion and
future works CSP6’
CSP5 CSP7 CSP8
42 / 86

43. Hybridization of
Idea 1 : Integrate split in GI
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Hybrid solving :
need a framework
Integrate split in GI
use the CP framework algorithm
Constraint propagation
Domain splitting
split and reduction at the same level
Theoretical model
for hybridization
work on union of CSP
CP + LS
Theoretical model
for hybridization
CP + GA
To an uniform
framework CP +
GA + LS
Conclusion and
future works
43 / 86

44. Hybridization of
Theoretical model for CSP solving (2)
complete and
incomplete
methods to solve
CSP
Reduction : by constraint propagation :
Tony Lambert
Solving CSP
Ω Ω’
Hybrid solving :
need a framework
Integrate split in GI
Constraint propagation
Domain splitting
Theoretical model
Ω
for hybridization
CSP1 CSP2 CSP3
CP + LS
Theoretical model
for hybridization
CP + GA
Constraint Propagation
To an uniform
framework CP +
GA + LS
Ω’
Conclusion and
CSP1 CSP2’ CSP3
future works
44 / 86

45. Hybridization of
Theoretical model for CSP solving (3)
complete and
incomplete
methods to solve
CSP
Tony Lambert
Reduction by domain splitting :
Solving CSP
Hybrid solving :
Ω Ω’
need a framework
Integrate split in GI
Constraint propagation
Domain splitting
Theoretical model
Ω
for hybridization
CSP1 CSP2 CSP3
CP + LS
Theoretical model
Split
for hybridization
CP + GA
To an uniform
framework CP +
Ω’
GA + LS
CSP1 CSP2’ CSP2’’ CSP2’’’ CSP3
Conclusion and
future works
45 / 86

46. Hybridization of
Theoretical model for CSP solving (1)
complete and
incomplete
methods to solve
Partial ordering :
CSP
Tony Lambert
Ω1
Solving CSP
Hybrid solving :
Application of a reduction :
need a framework
domain reduction function or
Ω2
Integrate split in GI
splitting function
Constraint propagation
Domain splitting
Theoretical model
for hybridization
Ω3
CP + LS
Set of CSP
Theoretical model
for hybridization
CP + GA
Ω4
To an uniform
framework CP +
GA + LS
Conclusion and
Ω5
future works
Terminates with a fixed point : set of solutions or inconsistent CSP
46 / 86

47. Hybridization of
complete and
incomplete
methods to solve
Solving CSP
1
CSP
Tony Lambert
Hybrid solving : need a framework
2
Solving CSP
Hybrid solving :
need a framework
Integrate split in GI
3
Integrate split in GI
Theoretical model
for hybridization
Theoretical model for hybridization CP + LS
4
CP + LS
LS as moves on a partial
ordering
Integrate samples for LS
Ordering σCSP
Theoretical model for hybridization CP + GA
5
A Generic Algorithm
Experimentation
Theoretical model
To an uniform framework CP + GA + LS
6
for hybridization
CP + GA
To an uniform
framework CP +
Conclusion and future works
7
GA + LS
Conclusion and
future works
47 / 86

48. Hybridization of
Our purpose :
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Hybrid solving :
Use of an existing theoretical model for CSP solving
need a framework
Definition of the solving process
Integrate split in GI
Theoretical model
for hybridization
CP + LS
Integrate :
LS as moves on a partial
ordering
Split
Integrate samples for LS
Ordering σCSP
A Generic Algorithm
LS
Experimentation
GA
Theoretical model
for hybridization
CP + GA
To an uniform
framework CP +
GA + LS
Conclusion and
future works
48 / 86

49. Hybridization of
Idea 2 : Integrate LS and CP
complete and
incomplete
methods to solve
CSP
Local Search
Tony Lambert
In theory : infinite
Solving CSP
In practice : number of iteration
Hybrid solving :
need a framework
Integrate split in GI
Characteristics of a LS path
Theoretical model
for hybridization
notion of solution
CP + LS
LS as moves on a partial
Operational (computational) point of view : path =
ordering
Integrate samples for LS
samples
Ordering σCSP
A Generic Algorithm
Experimentation
maximum length
Theoretical model
for hybridization
CP + GA
Goal :
To an uniform
framework CP +
LS ordering
GA + LS
integrate LS in CSP
Conclusion and
future works
49 / 86

50. Hybridization of
LS as moves on partial ordering
complete and
incomplete
methods to solve
CSP
Tony Lambert
p
Solving CSP
Function to pass from one LS 1
state to another
Hybrid solving :
need a framework
p
Integrate split in GI
2
Theoretical model
for hybridization
State of LS
CP + LS
p
LS as moves on a partial
ordering
3
Integrate samples for LS
Ordering σCSP
A Generic Algorithm
p
Experimentation
4
Theoretical model
for hybridization
CP + GA
p
To an uniform
5
framework CP +
GA + LS
Terminates with a fixed point : local minimum, maximum number of iteration
Conclusion and
future works
50 / 86

51. Hybridization of
Integrate samples for LS
complete and
incomplete
methods to solve
CSP
Tony Lambert
A sample :
Solving CSP
Depends on a CSP
Hybrid solving :
need a framework
Corresponds to a LS state
Integrate split in GI
Theoretical model
for hybridization
an SCSP contains (LS+domain reduction) :
CP + LS
LS as moves on a partial
A set of domains ;
ordering
Integrate samples for LS
Ordering σCSP
A (list of) sample for local search (a LS path).
A Generic Algorithm
Experimentation
Theoretical model
for hybridization
an σCSP contains (LS+domain reduction+split) :
CP + GA
A set of SCSP
To an uniform
framework CP +
GA + LS
Conclusion and
future works
51 / 86

52. Hybridization of
Theoretical model for hybridization (3)
complete and
incomplete
methods to solve
Ordering over σCSP
CSP
Tony Lambert
Ω1
Solving CSP
Hybrid solving :
Application of a reduction :
need a framework
Ω2
function :
Integrate split in GI
− domain reduction,
Theoretical model
− splitting,
for hybridization
− or LS step.
CP + LS
Ω3
LS as moves on a partial
ordering
Integrate samples for LS
Set of SCSP
Ordering σCSP
A Generic Algorithm
Ω4
Experimentation
Theoretical model
for hybridization
CP + GA
Ω5
To an uniform
framework CP +
GA + LS
A solution before the end of the process, given by LS
Conclusion and
Terminates with a fixed point : set of solutions or inconsistent SCSP
future works
52 / 86

53. Hybridization of
Algorithm
complete and
incomplete
methods to solve
CSP
Tony Lambert
Always the GI algorithm
Solving CSP
In the chaotic iteration framework
Hybrid solving :
need a framework
Finite domains (sets of SCSP)
Integrate split in GI
Theoretical model
Monotonic functions (LS move, domain reduction,
for hybridization
CP + LS
splitting)
LS as moves on a partial
ordering
Integrate samples for LS
Decreasing functions
Ordering σCSP
A Generic Algorithm
Experimentation
→ termination and computation of fixed point
Theoretical model
for hybridization
(solution of CP)
CP + GA
To an uniform
Practically : can stop before with a LS solution
framework CP +
GA + LS
Conclusion and
future works
53 / 86

54. Hybridization of
General process CP+LS
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Sets of
Constraint
Hybrid solving :
Application
SCSP
need a framework
of functions
Integrate split in GI
Theoretical model
for hybridization
CP + LS
LS as moves on a partial
ordering
Integrate samples for LS
Ordering σCSP
A Generic Algorithm
Experimentation
LS solution
Theoretical model
Global
for hybridization
CP + GA
Fixed point
To an uniform
framework CP +
GA + LS
Conclusion and
future works
54 / 86

55. Hybridization of
Strategies through reduction functions
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Hybrid solving :
need a framework
Reduction / Split / LS functions
Integrate split in GI
Theoretical model
Choose a / several SCPS to apply functions
for hybridization
CP + LS
LS as moves on a partial
Choose a / several domains (Prop. - Split)
ordering
Integrate samples for LS
Ordering σCSP
A Generic Algorithm
Tests : Random - Depth first - FC
Experimentation
Theoretical model
for hybridization
CP + GA
To an uniform
framework CP +
GA + LS
Conclusion and
future works
55 / 86

56. Hybridization of
Strategies : ratio LS-CP
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Ratio of LS and CP functions applied :
(red%,split%,ls%)
Hybrid solving :
need a framework
10% of split compared to reduction
Integrate split in GI
Theoretical model
CP alone : (90,10,0)
for hybridization
CP + LS
LS alone : (0,0,100)
LS as moves on a partial
ordering
Integrate samples for LS
Ordering σCSP
LS strategy : tabu
A Generic Algorithm
Experimentation
Theoretical model
Choose : LS, reduction, or split but keep the given
for hybridization
CP + GA
ratios
To an uniform
framework CP +
GA + LS
Conclusion and
future works
56 / 86

57. Hybridization of
Implementation
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Hybrid solving :
need a framework
Integrate split in GI
Theoretical model
C++
for hybridization
CP + LS
Generic Algorithm
LS as moves on a partial
ordering
Integrate samples for LS
choose function with percentages
Ordering σCSP
A Generic Algorithm
Experimentation
Theoretical model
for hybridization
CP + GA
To an uniform
framework CP +
GA + LS
Conclusion and
future works
57 / 86

58. Hybridization of
Langford number 4 2
complete and
incomplete
cooperation area
methods to solve
CSP
Tony Lambert
800
Solving CSP
Hybrid solving :
700
need a framework
Integrate split in GI
600
Theoretical model
for hybridization
CP + LS
Number of operations
500
LS as moves on a partial
ordering
Integrate samples for LS
400
Ordering σCSP
A Generic Algorithm
Experimentation
300
Theoretical model
for hybridization
CP + GA
200 Average
To an uniform
framework CP +
100
GA + LS
Conclusion and
future works 0
10 20 30 40 50 60 70 80 90 100
Propagation percentage
58 / 86

59. Hybridization of
SEND + MORE = MONEY
complete and
incomplete
cooperation area
methods to solve
CSP
Tony Lambert
1600
Solving CSP
Hybrid solving :
need a framework 1400
Integrate split in GI
Theoretical model 1200
for hybridization
CP + LS
Number of operations
LS as moves on a partial
1000
ordering
Integrate samples for LS
Ordering σCSP
A Generic Algorithm
800
Experimentation
Theoretical model
for hybridization
600
CP + GA
Average
To an uniform
framework CP + 400
GA + LS
Conclusion and
future works 200
10 20 30 40 50 60 70 80 90 100
Propagation percentage
59 / 86

60. Hybridization of
Ranges
complete and
incomplete
methods to solve
CSP
Tony Lambert
Problem S+M LN42 Zebra M. square Golomb
Solving CSP
Rate FC 70 - 80 15 - 25 60 - 70 30 - 45 30 - 40
Hybrid solving :
need a framework
Best range of propagation rate (α) to compute a solution
Integrate split in GI
Strategy Method S+M LN42 M. square Golomb
Theoretical model
Random LS 1638 383 3328 3442
for hybridization
CP+LS 1408 113 892 909
CP + LS
LS as moves on a partial
CP 3006 1680 1031 2170
ordering
Integrate samples for LS
D-First LS 1535 401 3145 3265
Ordering σCSP
A Generic Algorithm
CP+LS 396 95 814 815
Experimentation
CP 1515 746 936 1920
Theoretical model
for hybridization
FC LS 1635 393 3240 3585
CP + GA
CP+LS 22 192 570 622
To an uniform
CP 530 425 736 1126
framework CP +
GA + LS
Avg. nb. of operations to compute a first solution
Conclusion and
future works
60 / 86

61. Hybridization of
complete and
incomplete
methods to solve
Solving CSP
1
CSP
Tony Lambert
Hybrid solving : need a framework
2
Solving CSP
Hybrid solving :
need a framework
Integrate split in GI
3
Integrate split in GI
Theoretical model
for hybridization
Theoretical model for hybridization CP + LS
4
CP + LS
Theoretical model
for hybridization
CP + GA
Theoretical model for hybridization CP + GA
5
GA : moves in a partial
ordering
GA ordering
Integrate generations for
To an uniform framework CP + GA + LS
6
GA in CSPs
Algorithm
Application BACP (to
optimize loads of periods
Conclusion and future works
7
To an uniform
framework CP +
GA + LS
Conclusion and
future works
61 / 86

62. Hybridization of
Our purpose :
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Hybrid solving :
Use of an existing theoretical model for CSP solving
need a framework
Definition of the solving process
Integrate split in GI
Theoretical model
for hybridization
CP + LS
Integrate :
Theoretical model
Split
for hybridization
CP + GA
LS
GA : moves in a partial
ordering
GA ordering
GA
Integrate generations for
GA in CSPs
Algorithm
Application BACP (to
optimize loads of periods
To an uniform
framework CP +
GA + LS
Conclusion and
future works
62 / 86

63. Hybridization of
Idea 3 : GA = moves in a partial ordering
complete and
incomplete
methods to solve
CSP
Population
Tony Lambert
Génération k
Solving CSP
Hybrid solving :
Sélection
need a framework
Probabilité Pc
Probabilité Pm
Integrate split in GI
Theoretical model
for hybridization
p p1 p2
CP + LS
Theoretical model
Mutation
for hybridization
Croisement
CP + GA
GA : moves in a partial
ordering
p’
GA ordering
c1 c2
Integrate generations for
GA in CSPs
Algorithm
Evaluation
Application BACP (to
optimize loads of periods
To an uniform
Population
framework CP +
Génération k+1
GA + LS
Conclusion and
future works
63 / 86

64. Hybridization of
GA ordering
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Characteristics of a GA path
Hybrid solving :
need a framework
notion of solution
Integrate split in GI
maximum length
Theoretical model
for hybridization
Operational (computational) point of view : path =
CP + LS
generations
Theoretical model
for hybridization
CP + GA
GA : moves in a partial
Here a macroscopic point of view :
ordering
GA ordering
Integrate generations for
GA in CSPs
1 iteration = 1 generation
Algorithm
Application BACP (to
optimize loads of periods
To an uniform
framework CP +
GA + LS
Conclusion and
future works
64 / 86

65. Hybridization of
Integrate generations for GA in CSPs
complete and
incomplete
methods to solve
CSP
Tony Lambert
a generation :
Solving CSP
depends on a CSP
Hybrid solving :
need a framework
corresponds to a GA search state
Integrate split in GI
Theoretical model
for hybridization
an ogCSP contains (GA+domain reduction) :
CP + LS
a set of domains ;
Theoretical model
for hybridization
CP + GA
a (list of) population(s) for GA (a GA path).
GA : moves in a partial
ordering
GA ordering
Integrate generations for
an OGCSP contains (GA+domain reduction+split) :
GA in CSPs
Algorithm
Application BACP (to
a set of ogCSPs
optimize loads of periods
To an uniform
framework CP +
GA + LS
Conclusion and
future works
65 / 86

66. Hybridization of
Theoretical model for hybridization CP+GA
complete and
incomplete
Ordering over OGCSPs
methods to solve
CSP
Tony Lambert
Ω1
Solving CSP
Hybrid solving :
Application of a reduction:
need a framework
Ω2
function:
Integrate split in GI
− domain reduction
Theoretical model
− split
for hybridization
CP + LS
Ω3
− or GA.
Theoretical model
for hybridization
set of ogCSP
CP + GA
GA : moves in a partial
Ω4
ordering
GA ordering
Integrate generations for
GA in CSPs
Algorithm
Ω5
Application BACP (to
optimize loads of periods
To an uniform
framework CP +
an optimal solution given by GA before the end of the process
GA + LS
fixed point: set of solutions or inconsistency
Conclusion and
future works
66 / 86

67. Hybridization of
Algorithm
complete and
incomplete
methods to solve
CSP
Tony Lambert
Always the GI algorithm
Solving CSP
In the chaotic iteration framework
Hybrid solving :
need a framework
Finite domains (set of OGCSP)
Integrate split in GI
Theoretical model
Monotonic functions (GA move, domain reduction,
for hybridization
CP + LS
splitting)
Theoretical model
for hybridization
Decreasing functions
CP + GA
GA : moves in a partial
ordering
→ termination and computation of fixed point
GA ordering
Integrate generations for
GA in CSPs
(solution of CP)
Algorithm
Application BACP (to
optimize loads of periods
Practically : can stop before with an optimum for GA
To an uniform
framework CP +
GA + LS
Conclusion and
future works
67 / 86

68. Hybridization of
Application BACP (to optimize loads of
complete and
incomplete
periods)
methods to solve
CSP
Tony Lambert
Lectures Constraints
Solving CSP
a before b
g
a
e d cquot; quot;j
Hybrid solving :
need a framework
b
c hquot; quot;b
i h
Integrate split in GI
f j k iquot; quot;f
Theoretical model
fquot; quot;j
for hybridization
CP + LS
kquot; quot;c
Theoretical model
dquot; quot;e
for hybridization
Max
CP + GA
GA : moves in a partial
Min k
ordering
i e j
GA ordering
h
Integrate generations for
c f
GA in CSPs
Algorithm
b
d
Application BACP (to
g a
optimize loads of periods
To an uniform
Periods
framework CP +
GA + LS
Conclusion and
future works
68 / 86

69. Hybridization of
Experimentations
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Hybrid solving :
need a framework
Integrate split in GI
On optimization problems
Theoretical model
for hybridization
Here, ratios are (45,5,50)
CP + LS
Theoretical model
Mesure the real impact on the resolution
for hybridization
CP + GA
GA : moves in a partial
ordering
GA ordering
Integrate generations for
GA in CSPs
Algorithm
Application BACP (to
optimize loads of periods
To an uniform
framework CP +
GA + LS
Conclusion and
future works
69 / 86

70. Hybridization of
BACP8
complete and
incomplete
methods to solve
CSP
Tony Lambert
100
propagation
Solving CSP
genetic algorithm
Hybrid solving :
split
need a framework
80
Integrate split in GI
Theoretical model
60
for hybridization
range
CP + LS
Theoretical model
for hybridization
40
CP + GA
GA : moves in a partial
ordering
GA ordering
20
Integrate generations for
GA in CSPs
Algorithm
Application BACP (to
optimize loads of periods
0
To an uniform
25 24 23 22 21 20 19 18 17 16
framework CP +
GA + LS
evaluation
Conclusion and
future works
70 / 86

71. Hybridization of
BACP10
complete and
incomplete
methods to solve
CSP
Tony Lambert
100
propagation
Solving CSP
genetic algorithm
Hybrid solving :
split
need a framework
80
Integrate split in GI
Theoretical model
60
for hybridization
range
CP + LS
Theoretical model
for hybridization
40
CP + GA
GA : moves in a partial
ordering
GA ordering
20
Integrate generations for
GA in CSPs
Algorithm
Application BACP (to
optimize loads of periods
0
To an uniform
24 22 20 18 16 14 12
framework CP +
GA + LS
evaluation
Conclusion and
future works
71 / 86

72. Hybridization of
BACP12
complete and
incomplete
methods to solve
CSP
Tony Lambert
100
propagation
Solving CSP
genetic algorithm
Hybrid solving :
split
need a framework
80
Integrate split in GI
Theoretical model
60
for hybridization
range
CP + LS
Theoretical model
for hybridization
40
CP + GA
GA : moves in a partial
ordering
GA ordering
20
Integrate generations for
GA in CSPs
Algorithm
Application BACP (to
optimize loads of periods
0
To an uniform
25 24 23 22 21 20 19 18 17
framework CP +
GA + LS
evaluation
Conclusion and
future works
72 / 86

73. Hybridization of
complete and
incomplete
methods to solve
Solving CSP
1
CSP
Tony Lambert
Hybrid solving : need a framework
2
Solving CSP
Hybrid solving :
need a framework
Integrate split in GI
3
Integrate split in GI
Theoretical model
for hybridization
Theoretical model for hybridization CP + LS
4
CP + LS
Theoretical model
for hybridization
CP + GA
Theoretical model for hybridization CP + GA
5
To an uniform
framework CP +
GA + LS
To an uniform framework CP + GA + LS
6
Search Trees
Reduction fonctions
Conclusion and
future works
Conclusion and future works
7
73 / 86

74. Hybridization of
Motivation
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Hybrid solving :
need a framework
LS + CP : a search path
Integrate split in GI
GA + CP : a sequence of generations
Theoretical model
for hybridization
But : GA + CP + LS ?
CP + LS
Theoretical model
=⇒ Notion of sample = generation
for hybridization
CP + GA
=⇒ Need to consider sample and individual at the same
To an uniform
level : CSP level
framework CP +
GA + LS
Search Trees
Reduction fonctions
Conclusion and
future works
74 / 86

75. Hybridization of
Search Tree
complete and
incomplete
methods to solve
CSP
Tony Lambert
CSP1
propagation
Solving CSP
CSP1’
Hybrid solving :
split
need a framework
Integrate split in GI CSP2 CSP3
propagation
Theoretical model
for hybridization
CSP2’ CSP3
CP + LS
split
Theoretical model
CSP5
CSP4 CSP3
for hybridization
propagation
CP + GA
To an uniform CSP5
CSP4 CSP3’
framework CP +
split
GA + LS
Search Trees
CSP6
CSP4 CSP5 CSP7 CSP8
Reduction fonctions
propagation
Conclusion and
future works CSP6’
CSP5 CSP7 CSP8
75 / 86

76. Hybridization of
Example : Local Search
complete and
incomplete
methods to solve
CSP
Tony Lambert
CSP1 CSP2
Solving CSP
Hybrid solving :
Generate new samples
need a framework
CSP3 CSP5 CSP6 CSP7
CSP1 CSP2 CSP4 CSP8
Integrate split in GI
Theoretical model
for hybridization CSP3 CSP5 CSP6 CSP7
CSP1 CSP2 CSP4 CSP8
Chose a sample
CP + LS
CSP3 CSP5 CSP6 CSP7
CSP1 CSP2 CSP4 CSP8
Theoretical model
for hybridization
CP + GA
To an uniform
Generate new samples
framework CP + CSP7 CSP12
CSP10
CSP9 CSP11
CSP1 CSP6 CSP8
GA + LS
Search Trees
Reduction fonctions
CSP7 CSP12
CSP10
CSP9 CSP11
CSP1 CSP6 CSP8
Chose a sample
Conclusion and
future works
CSP7 CSP12
CSP10
CSP9 CSP11
CSP1 CSP6 CSP8
76 / 86

77. Hybridization of
Example : Genetic algorithms
complete and
incomplete
methods to solve
CSP
Tony Lambert
CSP3 CSP5
CSP1 CSP2 CSP4
Solving CSP
Hybrid solving :
Crossover
need a framework
CSP3 CSP5 CSP6
CSP1 CSP2 CSP4
Integrate split in GI
Theoretical model
for hybridization
CP + LS
Theoretical model
Mutation CSP3 CSP5 CSP6 CSP7
CSP1 CSP2 CSP4
for hybridization
CP + GA
To an uniform
framework CP +
GA + LS
Search Trees
Selection
Reduction fonctions
CSP3 CSP5 CSP6 CSP7
CSP1 CSP4
Conclusion and
future works
77 / 86

78. Hybridization of
Idea 4 : break up methods
complete and
incomplete
methods to solve
CSP
Tony Lambert
Basic components
Solving CSP
Hybrid solving :
reducing is a search component
need a framework
Integrate split in GI
splitting is a search component
Theoretical model
generating a neighborhood is a search component
for hybridization
CP + LS
moving to sample is a search component
Theoretical model
for hybridization
CP + GA
Basic behaviours
To an uniform
framework CP +
splitting and generating a neighborhood
GA + LS
Search Trees
reducing, selecting and moving to sample
Reduction fonctions
Conclusion and
future works
78 / 86

79. Hybridization of
Basic fonctions
complete and
incomplete
Sampling S :
methods to solve
CSP
P( X , C, P(D1 ) × · · · × P(Dn ) )
Tony Lambert
Solving CSP
→ P( X , C, P(D1 ) × · · · × P(Dn ) )
Hybrid solving :
need a framework
{φ1 , . . . , φn } → {φ1 , . . . , φn , φn+1 }
Integrate split in GI
Theoretical model
for hybridization
s.t. ∃φi with φi φn+1
CP + LS
Theoretical model
Reducing R :
for hybridization
CP + GA
P( X , C, P(D1 ) × · · · × P(Dn ) )
To an uniform
framework CP +
GA + LS
→ P( X , C, P(D1 ) × · · · × P(Dn ) )
Search Trees
Reduction fonctions
Conclusion and
{φ1 , . . . , φi , . . . , φn } → {φ1 , . . . , φi , . . . , φn }
future works
Where φi = ∅ or φ = X , C, Di and φ = X , C, Di s.t.
Di ⊆ Di .
79 / 86

80. Hybridization of
Reduction functions (1)
complete and
incomplete
methods to solve
CSP
Domain reduction (DR)
Tony Lambert
{φ1 , . . . , φi , . . . , φn } →DR {φ1 , . . . , φi , . . . , φn }
Solving CSP
Hybrid solving :
Where DR = Rm with m > 0
need a framework
Integrate split in GI
Split (SP)
Theoretical model
for hybridization
CP + LS
Theoretical model
{φ1 , . . . , φi , . . . , φn } →SP {φ1 , . . . , φ1 , . . . , φm , . . . , φn }
for hybridization
i i
CP + GA
To an uniform
Where SP = S m R
framework CP +
GA + LS
Search Trees
Local Search (LS)
Reduction fonctions
Conclusion and
future works
{φ1 , . . . , φi , . . . , φn } →LS {φ1 , . . . , φi , . . . , φn }
Where LS = S m Rm−1
80 / 86

81. Hybridization of
Reduction functions (2) Genetic Algorithms
complete and
incomplete
methods to solve
CSP
Crossover
Tony Lambert
Solving CSP
{φ1 , . . . , φn } →CR {φ1 , . . . , φn , φn+1 }
Hybrid solving :
need a framework
Where CR = S
Integrate split in GI
Theoretical model
Mutation
for hybridization
CP + LS
{φ1 , . . . , φi , . . . , φn } →MU {φ1 , . . . , φi , . . . , φn }
Theoretical model
for hybridization
CP + GA
Where MU = SR
To an uniform
framework CP +
GA + LS
Selection
Search Trees
Reduction fonctions
{φ1 , . . . , φn } →SE {φ1 , . . . , φn }
Conclusion and
future works
Where SE = Rm .
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82. Hybridization of
Properties
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Hybrid solving :
need a framework
Integrate split in GI
a strategy is a sequence of words
Theoretical model
∈ {DR, SP, LS, SE, CR, MU}
for hybridization
CP + LS
is it finite sequences ?
Theoretical model
for hybridization
need conditions to avoid loops
CP + GA
To an uniform
framework CP +
GA + LS
Search Trees
Reduction fonctions
Conclusion and
future works
82 / 86

83. Hybridization of
complete and
incomplete
methods to solve
Solving CSP
1
CSP
Tony Lambert
Hybrid solving : need a framework
2
Solving CSP
Hybrid solving :
need a framework
Integrate split in GI
3
Integrate split in GI
Theoretical model
for hybridization
Theoretical model for hybridization CP + LS
4
CP + LS
Theoretical model
for hybridization
CP + GA
Theoretical model for hybridization CP + GA
5
To an uniform
framework CP +
GA + LS
To an uniform framework CP + GA + LS
6
Conclusion and
future works
Assessment
Conclusion and future works
Future
7
83 / 86

84. Hybridization of
Conclusion
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Hybrid solving :
need a framework
A generic model for hybridizing complete (CP) and
Integrate split in GI
incomplete (LS and GA) methods
Theoretical model
for hybridization
Implementation of modules working on the same
CP + LS
structure
Theoretical model
for hybridization
CP + GA
Complementarity of methods : hybridization
To an uniform
framework CP +
GA + LS
Conclusion and
future works
Assessment
Future
84 / 86

85. Hybridization of
Future
complete and
incomplete
methods to solve
CSP
Tony Lambert
Solving CSP
Hybrid solving :
synergies between :
need a framework
complete + incomplete methods,
Integrate split in GI
CP + AI + OR + . . .
Theoretical model
for hybridization
learning strategies, rules based system with a
CP + LS
language
Theoretical model
for hybridization
CP + GA
providing more tools in a generic environment
To an uniform
Design of strategies
framework CP +
GA + LS
Conclusion and
future works
Assessment
Future
85 / 86

86. Hybridization of
complete and
incomplete
methods to solve
CSP
Hybridization of complete and
Tony Lambert
incomplete methods to solve CSP
Solving CSP
Hybrid solving :
need a framework
Integrate split in GI
Tony Lambert
Theoretical model
for hybridization
CP + LS
LERIA, Angers, France
Theoretical model
for hybridization LINA, Nantes, France
CP + GA
To an uniform
October 27th, 2006
framework CP +
GA + LS
Conclusion and
future works
Rapporteurs : François FAGES Directeur de Recherche, INRIA Rocquencourt
Assessment
Bertrand NEVEU, Ingénieur en Chef des Ponts et Chaussées, HDR
Future
Examinateurs : Steven PRESTWICH, Associate Professor, University College Cork
Jin-Kao HAO, Professeur à l’Université d’Angers
Directeurs de thèse : Éric MONFROY, Professeur à l’Université de Nantes
Frédéric SAUBION, Professeur à l’Université d’Angers
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