Assumption-based Pruning in Conditional CSP

IBM Labs in Haifa © 2005 IBM Corporation

Assumption-based Pruning in Conditional CSP

Felix Geller and Michael Veksler

2 © 2005 IBM Corporation

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Outline

Conditional CSP (Mittal and Falkenhainer) Conditional CSP solution methods Activity CSP formulation MAC-based algorithm for ACSP Activity disjunction and implication Questions


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Describes CSP where not all variables are required to participate in a solution (V, VI, D,CC,CA)

Conditional CSP (Mittal and Falkenhainer)

The assigned variables are active variables A compatibility constraint is relevant if all its variables are active

vCincl

IVVv \Activity of a variable

is determined by activity constraint

sIs VVVV ,Given an assignment of values to

vCincl

An activity constraint: is relevant if C is relevant

An activity constraint is satisfied if v is active or C is not satisfied In a solution all relevant constraints have to be satisfied

Minimality requirement


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Existing approaches to solving Conditional CSPs

Reformulation Working directly on Conditional CSP representation

Main idea: maintain list of active variables and check/propagate constraints after their variables become active

[Mittal,Falkenhainer]: Backtrack search with conflicts recording [Gelle,Faltings]: At first determine variable activity status; for fixed

variable activity status CondCSP turns into CSP. [Sabin,Freuder,Wallace], [Gelle,Sabin]: invoke consistency techniques

(CondFC, CondMAC) directly on Conditional CSP representation Propagate constraints once their variables are active CondMAC proved to be superior over CondBT and CondFC

The limitation: constraint propagation has to wait until variables become active


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Activity CSP formulation (V, VI, VA,D,C,A)

Variable is active if either or is trueIVvv )(vA

Constraint is relevant if all its variables are active An assignment is a solution if all relevant constraints are

satisfied

AVvA )(IVVv \ A - activation conditions:

Each variable is dominated by some

IA VV Explicit activity variables

sIs VVVV ,Given an assignment of values to


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Early conflict detection: ey = true D(x)= D(y) = {0,…,4}ez = true D(x)= D(z) = {5,…,9}D(y) ∩ D(z) = Ø

ey ez = false Information flow from conditional variables to unconditional variables:

ey = true D(x)= D(y) = {0,…,4}ez = true D(x)= D(z) = {5,…,9}ey ez

D(X)= {0…9}

An ACSP problem

{true,false}

ey

{true,false}ez

{0,…,4}

y

{5,…,9}

z

{0,…,100}

x

= =

V


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Enabling early constraint propagation

Allow constraint propagation to modify variable domains before the constraint variables become active

ey ez

{0,…,4}

y

{5,…,9}

z

z[ey,ez]

{5,…,9}

Activity set

ezey

y[ey,ez]

{0,…,4}

Shadows


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Assumption-based decomposition

{0,…4} {5,…,9}

{true,false}

ey

{true,false}ez

y z

{0,…,100}

x

= =

x

{0,…,100}

ey ez

y

x[ey]

{0,…,4}

{0,…,100}

=

z

x[ez]

{5,…,9}

{0,…,100}

=

Collect assumptions (activity sets)

Create shadows

Group shadows and constraints into CSP sub-problems


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Shadow synchronization

y

x[ey]

z

x[ez]

{0,…,4} {5,…,9}

{0,…,100}

{0,…,100}

x

{0,…,100}

ey ez

{true,false} {true,false}

{0,…,4} {5,…,9}

ey ez = false

{3}


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Shadow synchronization

Two shadows of the same variable: x[AS1] and x[AS2]

If under the current partial solution, AS2 implies AS1 (AS1 is weaker than AS2 )

D(x[AS2]) D(x[AS2]) ∩ D(x[AS1]

If an empty domain results, some activity var in AS2 is false

If AS1 and AS2 are incomparable, but D(x[AS1]) ∩ D(x[AS2]) = Ø

Conclude that some var in AS1 U AS2 is false


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Preprocessing While there are unassigned variables

Select a variable and instantiate it Propagate constraints

While constraint queue is not empty Select and propagate a constraint Synchronize shadows Update constraint queue

Backtrack if failed

AMAC - Putting all things together

Compute constraints activity sets

Create shadows

Substitute constraints parameters with suitable shadows

Select an active variable

Activity inference Handle synchronization and

constraint propagation failures Combine activity deductions to

determine value of activity variables


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Experimental results - setting

Random Conditional CSP generator similar to the generator in [Sabin,Freuder,Wallace]

CSP over N variables with binary (compatibility) constraints Problem parameters:

))1|*(||/(|||*2 VVCdcCompatibility density

Compatibility satisfiability ||*||/# jic DDtupless

||/|\| VVVd Ia Activity density

||/|| iia DSs j

incl

ii vSv Activation satisfiability (for )

N=48, |VI|=12, sa=0.75, dc=0.15


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Experimental results – 1 activity cluster


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Experimental results – 4 activity clusters


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Experimental results – 36 independent activation conditions


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Experimental results – effect of clustering (sc=0.25)


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Activity disjunction

{0,…,100}

y

x[ey]

{0,…,4}

{0,…,4}

x

z

x[ez]

{5,…,9}

{5,…,9}

ey ez

V

{true,false} {true,false}

{12}{0,…,9}


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Activity implication

Implication constraints between activity variables induce equivalence between activity sets

ey ez {ey} ~ {ey, ez}

Equivalence allows to reduce the number of shadow variables


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Summary

Introduced Activity CSP – a variant of Conditional CSP formulation ACSP directly supports full constraint propagation

Activity MAC checks several activity assumptions in parallel Infers that an assumption is false if the constraint propagation fails

ACSP naturally captures clustering and activity disjunction Interpret activity disjunction and implication in terms of activity sets

and shadows operations Further improve propagation and information flow

AMAC substantially reduces number of backtracks and solution time on random benchmarks On easy problems adds overhead of shadow/assumption

manipulation


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Assumption-based decomposition Constraint Activity set

C1(x,ey)

C2(x,y)

C3(y,ez)

C4(x,z)

C5(y,z)

ey ez

y

zx

ey

ey

ez

ezey

x

ey ez

ez[ey]

x[ey]

y

zx[ez]

y[ey,ez]

z[ey,ez]

Assumption-based Pruning in Conditional CSP

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