An Improved Algorithm forMaintaining

Arc Consistency in DynamicConstraint Satisfaction Problems

Pavel SurynekCzech Technical UniversityPrague

pavel.surynek@seznam.cz

Roman BartákCharles UniversityPrague

roman.bartak@mff.cuni.cz

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.

Existing approaches to DCSPs

Search for robust solution the solution is still valid for the problem after

a 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

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

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

AC|DC (filtration based on AC-3) simple low memory consumption slow

AC3.1|DC (filtration based on AC-3.1) fast larger memory consumption

uses additionaldata structures

no additionaldata structures

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)

A new algorithm AC|DC-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 AC3.1|DC-2i

B: 2 3 42/D6 3/D9

Constraint addition by AC|DC-2i

E:

B=D (1)

C=D (3)

A<C (2)C≠E (4)

order number when the constraint is added

A: 2 3 4 3

justification for value removal VariableTime

4/C2

Justifications and removal times are recorded during addition of constraints by AC-3

C: 1 2 3 41/A3 2/A4 3/E7

D: 1 2 3 41/B1 2/C5 3/C8

B: 2 42/D6 3/D9

Constraint retraction by AC|DC-2i

E:

B=D (1)

C=D (3)

A<C (2)C≠E (4)

D: 2 41/B1

C: 1 2 4

A: 2 3 4 34/C2

1/A3 2/A4

2/C5

3/E7

3/C8

Constraint A<C is removed from the problem

Initialization: restore values deleted when propagating A<CInitialization: restore values deleted when propagating A<C

Constraint A<C is removed from the problem

Propagation: domain extensions are propagatedPropagation: domain extensions are propagated

Filtration: remove inconsistent values (re-establish AC)Filtration: remove inconsistent values (re-establish AC)

1/D10

Experimental resultsRuntime comparison of retraction from a consistent state

0

10

20

30

40

50

60

87 87.2 87.4 87.6 87.8 88 88.2 88.4 88.6 88.8 89Tightness in percents (100*p2)

Tim

e in

sec

onds

AC|DC

AC3.1|DCDnAC-6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

87 87.2 87.4 87.6 87.8 88 88.2 88.4 88.6 88.8 89Tightness in percents (100*p2)

Tim

e in

sec

onds

DnAC-6AC|DC-2iAC3.1|DC-2i

RCSP(100,50,0.5,p2)

Experimental resultsRuntime of constraint addition

0

2

4

6

8

10

12

14

16

18

20

87 87.2 87.4 87.6 87.8 88 88.2 88.4 88.6 88.8 89Tightness in percents (100*p2)

Tim

e in

sec

onds

AC|DC AC|DC-2iDnAC-6 AC3.1|DCAC3.1|DC-2i

RCSP(100,50,0.5,p2)

Experimental resultsMemory consumption

Domain size (d) 20 30 40 50 60 70 80 90

Tightness of constraints

(100*p2)71% 79% 84% 87% 89% 90% 91% 92%

DnAC-6 2MB 4MB 6MB 7MB 9MB 10MB 12MB 13MB

AC|DC <1MB <1MB <1MB <1MB <1MB <1MB <1MB <1MB

AC3.1|DC 2MB 3MB 5MB 5MB 7MB 7MB 9MB 10MB

AC|DC-2i <1MB <1MB <1MB <1MB <1MB <1MB <1MB <1MB

AC3.1|DC-2i 2MB 3MB 5MB 5MB 7MB 7MB 9MB 10MB

RCSP(100,d,0.5,p2)

Conclusions

New algorithm for maintaining arc consistency AC|DC-2i practical time of constraint retraction better than

DnAC-6 (so far fastest) low memory consumption (like simple AC|DC)

Optionally use AC-3.1 AC3.1|DC-2i improves time of constraint addition larger memory consumption