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An Elegant and Efficient An Elegant and Efficient Implementation of Implementation of

Russian Dolls Search Russian Dolls Search for Variable Weighted CSPfor Variable Weighted CSP

T. Benoist, M. Lemaître

tbenoist@bouygues.com

Bouygues e-lab, Paris, FranceONERA, Toulouse, France

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Variable Weighted CSPVariable Weighted CSP• Definition:

– Additive WCSP where only unary constraints (Xi=1) are weighted

– maximize Ω = ∑wiXi

– Subject to: Xi∈0,1 + selection constraints• Some VWCSPs:

– knapsack problems, – «soft scheduling »– prize-collecting TSP– …

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Example: Select & Schedule

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10 candidate photos, each with gain wiSelect the subset (selection variables Xi)

of higher gain (Ω=∑wiXi)that can be scheduled (variables Ti) without

violating transition times ti→j constraints.

Simplified version of SRSS [Verfaillie & Lemaître CP’01]

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Example: Select & Schedule10 candidate photos, each with gain wi

Select the subset (selection variables Xi)of higher gain (Ω=∑wiXi)that can be scheduled (variables Ti) without

violating transition times ti→j constraints.10

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Simplified version of SRSS [Verfaillie & Lemaître CP’01]

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Russian Dolls Search• Static order on selection variables• Successive resolutions of nested

sub problems

Rds10=max(P10)

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814

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Russian Dolls Search• Static order on selection variables• Successive resolutions of nested

sub problemsRds9=max(P9)

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814

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Russian Dolls Search

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Rds8=max(P8)

and so on…

• Static order on selection variables• Successive resolutions of nested

sub problems

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Russian Dolls SearchRds1=max(P1)=max(P)

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RDS upper boundDuring resolution of the last problem (P1)(once P10,P9,…,P2 have been solved)

Initial upper bound: Ω ≤ w1+Rds2

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10Rds2

1w1

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RDS upper bound

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814

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RDS upper bound

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5Constraints propagation

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RDS upper bound

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frontier

Constraints propagation

Rds6

w4

Ub=w1+ w2+ w4+Rds6

1w1

2w2

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RDS cost-based filteringVariable fixing rule:

If w1+ w2+ Rds6 ≤ currentBest then X4=1

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10Rds6w1

4w2

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RDS cost-based filteringVariable fixing rule:

If w1+ w2+ Rds6 ≤ currentBest then X4=1

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10Rds6w1

4w4

w2

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LightRDS• Objective function:

• Ω = w1X1+ ... + w5X5+ Ω6

• Ω6= w6X6+ ... + w10X10 with Ω6≤Rds6

• RDS filtering is naturally performed by these linear constraints

Ligh

tRDS

Ω = Ω1

Ωi = wiXi+ Ωi+1

Ω10= w10X10

Declarative implementation

Adding 10 variables and 10 ternary constraints

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LightRDS filtering is strictly strongerLi

ghtR

DS

This is equivalent to using all frontiers simultaneously

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LightRDS filtering is strictly stronger

If Rds4<Rds6+w4 then frontier 4 would produce a better bound:

w1+ w2+ Rds4 < w1+ w2+ w4+Rds6

Ligh

tRDS

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10Ω4

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LightRDS filtering is strictly stronger

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If Rds4<Rds6+w4 then frontier 4 would produce a better bound:

w1+ w2+ Rds4 < w1+ w2+ w4+Rds6

And possibly better filtering:

Ligh

tRDS

Ω4

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Ligh

tRDS

Control mechanism

Encapsulated in a special root choice point

Restrict to problem P10

X1=X2=….=X9=0

setMax(Ω10,currentBest)

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Ligh

tRDS

Control mechanism

Encapsulated in a special root choice point

P10 P9 P8P1

………..

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Computational Results

Ligh

tRDS

• Satellite planning problem: ftp://ftp.cert.fr/pub/DCSD/CD/lemaitre/Choco/bep/

• CHOCO model

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80 100 120 140 160 180 200

CP

U T

ime

(sec

onds

)

nbCandidates

LightRDSRDS

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200

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80 100 120 140 160 180 200T

ree

sear

ch s

ize

(# n

odes

/100

0)nbCandidates

LightRDSRDS

Number of nodes and CPU time are divided by two

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NOT extensible to WCSP

∑wcXc

• Forward Checking (FC)= ∑wcXc + ∑wcXc + ∑wcXc+ … + ∑wcXcc∈F* c∈F1 c∈F2 c∈F0

Instantiated constraints

Constraints whose only non

instantiated variable is V1

Constraints whose only non

instantiated variable is V2

Remaining constraints

RDS bound of the largest subproblem ⊆ F0

Bounded by

Min ∑wcXc

Bounded by

domain(V1) c∈F1

dynamic partition of constraintsstatic reformulation of the objective function seems impossible

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Conclusion• LightRDS is more efficient• LightRDS is simple:

– Declarative implementation– No dedicated filtering to program– No frontier to manage

• LightRDS can be tested in a few minutes– on Variable Weighted CSPs– when the constraint graph has a small bandwidth