Lambert Sac2005

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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Outline

• Hybridization
• Constraint propagation
• Local search
• Hybrid model
• Some results
• Conclusion

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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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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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Hybridization : getting the best of both methods

• But, generally :
• Ad-hoc systems
• Master-slaves approaches
• Coarse grain cooperation

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• Ad-hoc systems
• Idea :
• Fine grain control
• More strategies

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• Ad-hoc systems
• Idea :
• Fine grain control
• More strategies
• Technique :
• Decomposing solvers into basic functions
• Adapting chaotic iterations for hybrid solving

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CSP (Constraint Satisfaction Problem)

C2
Y
C1 U
Variables
X
C4 C5
Constraints
Z T
C3

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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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Solving CSP with a complete method

Constraint
Propagation

Enumeration
Splitting

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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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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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Propagation

Local consistency

Propagation

inconsistency

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Constraint propagation and domain splitting

Propagation

Splitting

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Constraint propagation and domain splitting

Propagation

Splitting
Propagation

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Abstract Model K.R. Apt [CP99]

Abstraction
Reduction functions
Constraint

Application
of functions
CSP
Fixed point
Partial Ordering

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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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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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Local search

• Search space : set of possible conﬁgurations
• Tools : neighborhood and evaluation function

Set of
possible
configurations

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Local search


Neighborhood
of s Set of
possible
configurations

s3
s4
s6
s1 s2
s5

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Local search


Neighborhood
of s Set of
possible
configurations

s3
s4
s6
s1 s2
s5

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Local search


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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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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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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Reduction : by constraint propagation

Ω Ω’

Ω CSP1 CSP2 CSP3


Ω CSP1 CSP2’ CSP3

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Reduction : by domain splitting

Ω Ω’

Ω CSP1 CSP2 CSP3

Split

Ω CSP1 CSP2’ CSP2’’ CSP2’’’ CSP3

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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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LS Ordering

Characteristics of a LS path
• Notion of Solution
• Maximum Length

Operational (computational) point of view : path = samples

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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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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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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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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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General Process

Application
of functions
Sets of
Constraint
SCSP

LS solution
Global
Fixed point

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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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Examples

• SEND + MORE = MONEY
• Zebra
• Golomb
• Magic square
• Langford Number

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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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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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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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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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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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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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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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Lambert Sac2005

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