An Improved Algorithm for Maintaining Arc Consistency in Dynamic Constraint Satisfaction Problems

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An Improved Algorithm forMaintainingArc
Consistency in DynamicConstraint Satisfaction
Problems
Pavel SurynekCzech Technical UniversityPraguep
avel.surynek_at_seznam.cz
Roman BartákCharles UniversityPragueroman.bart
ak_at_mff.cuni.cz
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Problem area

• Real world dynamic world
• practical problems change continuously
• difficult to capture by static formulation
• solution must reflect that changes
• Dynamic CSP
• A CSP from which constraints can be retracted or
  to which constraints can be added in arbitrary
  order.

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Existing approaches to DCSPs

• Search for robust solution
• the solution is still valid for the problem
  aftera small change
• Reconstruction of the solution
• the solution is locally repaired after a change
• Minimal perturbation problems
• solutions minimizing that local repair
• Reusing the reasoning process
• our approach maintaining arc consistency

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Why dynamic arc consistency ?

• Arc consistency simplifies the problem
• Interactive problems
• Interactive preparation of consistent a problem
• peptide synthesis
• determination of RNA structures
• timetabling
• Search algorithms
• Decision addition of a constraint
• Undoing decision retraction of a constraint

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Current situation

• DnAC-4 (filtration based on AC-4)
• quite fast
• large memory consumption
• DnAC-6 (filtration based on AC-6)
• so far fastest algorithm for maintaining AC
• large memory consumption
• complicated data structures
• ACDC (filtration based on AC-3)
• simple
• low memory consumption
• slow
• AC3.1DC (filtration based on AC-3.1)
• fast
• larger memory consumption

uses additionaldata structures
no additionaldata structures
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3 phases of constraint retraction

• Initialization phase restores
• values deleted by the retracted constraint from
  domains of its variables when propagating through
  it for the first time
• Propagation phase restores
• values that can be added due to previous domain
  extensions (like reverted AC)
• Filtration phase removes
• inconsistent values (standard AC)

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A new algorithm ACDC-2i

• Record information during addition of constraints
  via AC-3
• justifications (like DnACs, neighbor in which
  lost all supports) and value removal time
• Restore only most promising values
• use removal times and justifications to identify
  values to restore
• new support in justification variable
• that was deleted before restored value
• Optionally use AC-3.1 ? AC3.1DC-2i

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Constraint addition by ACDC-2i
Justifications and removal times are recorded
during addition of constraints by AC-3
order number when the constraint is added
justification for value removal VariableTime
BD (1)
B
2
3
4
2/D6
3/D9
D
1
2
3
4
1/B1
2/C5
3/C8
CD (3)
C
1
2
3
4
1/A3
2/A4
3/E7
AltC (2)
C?E (4)
E
A
2
3
4
3
4/C2
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Constraint retraction by ACDC-2i
Constraint AltC is removed from the problem
Constraint AltC is removed from the problem
Initialization restore values deleted when
propagating AltC
Initialization restore values deleted when
propagating AltC
Propagation domain extensions are propagated
Propagation domain extensions are propagated
Filtration remove inconsistent values
(re-establish AC)
Filtration remove inconsistent values
(re-establish AC)
BD (1)
3/D9
B
2
4
2/D6
D
2
4
3/C8
1/B1
2/C5
CD (3)
1/A3
C
1
2
4
2/A4
3/E7
1/D10
AltC (2)
C?E (4)
E
A
2
3
4
3
4/C2
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Experimental resultsRuntime comparison of
retraction from a consistent state
RCSP(100,50,0.5,p2)
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Experimental resultsRuntime of constraint
addition
RCSP(100,50,0.5,p2)
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Experimental resultsMemory consumption
RCSP(100,d,0.5,p2)
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Conclusions

• New algorithm for maintaining arc consistency
  ACDC-2i
• practical time of constraint retraction better
  than DnAC-6 (so far fastest)
• low memory consumption (like simple ACDC)
• Optionally use AC-3.1 ? AC3.1DC-2i
• improves time of constraint addition
• larger memory consumption