University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln CSE Conference and Workshop Papers Computer Science and Engineering, Department of 2011 Reformulating the Dual Graphs of CSPs to Improve the Performance of RNIC Robert J. Woodward University of Nebraska-Lincoln, [email protected] Shant Karakashian University of Nebraska-Lincoln, [email protected] Berthe Y. Choueiry University of Nebraska-Lincoln, [email protected] Christian Bessiere University of Montpellier, France, [email protected] Follow this and additional works at: hp://digitalcommons.unl.edu/cseconfwork Part of the Computer Sciences Commons is Article is brought to you for free and open access by the Computer Science and Engineering, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in CSE Conference and Workshop Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Woodward, Robert J.; Karakashian, Shant; Choueiry, Berthe Y.; and Bessiere, Christian, "Reformulating the Dual Graphs of CSPs to Improve the Performance of RNIC" (2011). CSE Conference and Workshop Papers. 185. hp://digitalcommons.unl.edu/cseconfwork/185
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University of Nebraska - LincolnDigitalCommons@University of Nebraska - Lincoln
CSE Conference and Workshop Papers Computer Science and Engineering, Department of
2011
Reformulating the Dual Graphs of CSPs toImprove the Performance of RNICRobert J. WoodwardUniversity of Nebraska-Lincoln, [email protected]
Shant KarakashianUniversity of Nebraska-Lincoln, [email protected]
Berthe Y. ChoueiryUniversity of Nebraska-Lincoln, [email protected]
Christian BessiereUniversity of Montpellier, France, [email protected]
Follow this and additional works at: http://digitalcommons.unl.edu/cseconfwork
Part of the Computer Sciences Commons
This Article is brought to you for free and open access by the Computer Science and Engineering, Department of at DigitalCommons@University ofNebraska - Lincoln. It has been accepted for inclusion in CSE Conference and Workshop Papers by an authorized administrator ofDigitalCommons@University of Nebraska - Lincoln.
Woodward, Robert J.; Karakashian, Shant; Choueiry, Berthe Y.; and Bessiere, Christian, "Reformulating the Dual Graphs of CSPs toImprove the Performance of RNIC" (2011). CSE Conference and Workshop Papers. 185.http://digitalcommons.unl.edu/cseconfwork/185
Constraint Systems Laboratory
R.J. Woodward, S. Karakashian, B.Y. Choueiry & C. Bessiere
Constraint Systems Laboratory, University of Nebraska-Lincoln
LIRMM-CNRS, University of Montpellier
Reformulating the Dual Graphs of CSPs
to Improve the Performance of RNIC
Acknowledgements
• Elizabeth Claassen and David B. Marx of the Department of Statistics @ UNL
• Experiments conducted at UNL’s Holland Computing Center
• Robert Woodward supported by a B.M. Goldwater Scholarship and NSF Graduate Research Fellowship
• NSF Grant No. RI-111795
1/23/2012 SARA 2011 1
Constraint Systems Laboratory
Outline • Introduction
• Relational Neighborhood Inverse Consistency
– Property & algorithm
• Reformulating the Dual Graph by
1. Removing redundant edges, yields property wRNIC
2. Triangulation, yields property triRNIC
• Selection strategy: four alternative dual graphs
• Experimental Results
• Conclusion
1/23/2012 SARA 2011 2
Constraint Systems Laboratory
Constraint Satisfaction Problem
• CSP
– Variables, Domains
– Constraints: binary / non-binary
• Representation
– Hypergraph
– Dual graph
• Solved with
– Search
– Enforcing consistency
1/23/2012 SARA 2011 3
R4
BCD
ABDE
CF
EF AB
R3 R1
R2
C
F
E
BD
AB
D AD
A AD B
R5
R6
R3
A B
C D
E
F
R1
R4
R2 R5
R6
Hypergraph
Dual graph
• Warning
– Consistency properties vs. algorithms
Constraint Systems Laboratory
Neighborhood Inverse Consistency [Freuder+ 96]
• Property
– Defined for binary CSPs
– Every value can be extended to a
solution in its variable’s neighborhood
• Algorithm
⧾No space overhead
⧾Adapts to the connectivity
⧿Not effective on sparse problems
⧿To costly on dense problems
1/23/2012 SARA 2011 4
0,1,2
0,1,2
0,1,2
0,1,2
R0 R1 R3
R2
R4 A
B
C
D
R3
A B
C D
E
F
R1
R4
R2 R5
R6
• Non-binary CSPs?
⧿Neighborhoods likely too large
Constraint Systems Laboratory
Relational NIC [Woodward+ AAAI11]
1/23/2012 SARA 2011 5
• Property
– Defined for dual graph
– Every tuple can be extended to a
solution in its relation’s
neighborhood
• Algorithm
– Operates on dual graph
– … filter relations (not domains!)
R4
BCD
ABDE
CF
EF AB
R3 R1
R2
C
F
E
BD
AB
D AD
A AD B
R5
R6
R3
A B
C D
E
F
R1
R4
R2 R5
Hypergraph
Dual graph
• Domain filtering
– Property: RNIC+DF
– Algorithm: Projection
Constraint Systems Laboratory
• High density
– Large neighborhoods
– Higher cost of RNIC
• Minimal dual graph
– Equivalent CSP
– Computed efficiently [Janssen+ 89]
• Run algorithm on a minimal dual graph
⧾Smaller neighborhoods, solution set not affected
⧿wRNIC: a strictly weaker property
Reformulation: Removing Redundant Edges
1/23/2012 SARA 2011 6
R4
BCD
ABDE
CF
EF AB
R3 R1
R2
C
F
E
BD
AB
D AD
A AD B
R5
R6
dGo = 60%
dGw = 40%
wRNIC RNIC
Constraint Systems Laboratory
Reformulation: Triangulation
• Cycles of length ≥ 4
– Hampers propagation
• Triangulating dual graph
– Equivalent CSP
– Min-fill heuristic
• Run algorithm on a triangulated dual
graph
⧾Created loops enhance propagation
– triRNIC: a strictly stronger property
1/23/2012 SARA 2011 7
R4
BCD
ABDE
CF
EF AB
R3 R1
R2
C
F
E
BD
AB
D AD
A AD B
R5
R6
dGo = 60%
dGtri = 67%
wRNIC RNIC triRNIC
Constraint Systems Laboratory
Reformulation: RR & Triangulation
• Fixing the dual graph
– RR copes with high density
– Triangulation boosts propagation
• RR+Tri
– Both operate locally
– Are complementary, do not ‘clash’
• Run algorithm on a RR+tri dual
graph
– CSP solution set is not affected
– wtriRNIC is not comparable with RNIC
1/23/2012 SARA 2011 8
R4
BCD
ABDE
CF
EF AB
R3 R1
R2
C
F
E
BD
AB
D AD
A AD B
R5
R6
dGo = 60%
dGwtri = 47% R4
BCD
ABDE
CF
EF AB
R3 R1
R2
C
F
E
BD
AB
D AD
A AD B
R5
R6
wRNIC RNIC
wtriRNIC triRNIC
Constraint Systems Laboratory
Selection Strategy: Which? When?
• Density ≥ 15% is too dense
– Remove redundant edges
• Triangulation increases density no more than two fold
– Reformulate by triangulation
• Each reformulation executed at most once
1/23/2012 SARA 2011 9
No
Yes No Yes
Yes
No
dGo ≥ 15%
dGtri ≤ 2 dGo dGwtri ≤ 2 dGw
Go Gwtri Gw Gtri
Start
Constraint Systems Laboratory
Experimental Results • Statistical analysis on CP benchmarks
• Time: Censored data calculated mean
• R: Censored data rank based on
probability of survival data analysis
• S: Equivalence classes based on CPU
1/23/2012 SARA 2011 10
Algorithm Time #F R S #C SB #BF 169 instances: aim-100,aim-200,lexVg,modifiedRenault,ssa
wR(*,2)C 944924 52 3 A 138 B 79
wR(*,3)C 925004 8 4 B 134 B 92
wR(*,4)C 1161261 2 5 B 132 B 108
GAC 1711511 83 7 C 119 C 33
RNIC 6161391 19 8 C 100 C 66
triRNIC 3017169 9 9 C 84 C 80
wRNIC 1184844 8 6 B 131 B 84
wtriRNIC 937904 3 2 B 144 B 129
selRNIC 751586 17 1 A 159 A 142
• SB: Equivalence classes based on
completion
• #C: Number of instances completed
• #F: Number of instancesfastest
• #BF: # instances solved backtrack free
Constraint Systems Laboratory
Conclusions
• Contributions
– Algorithm
• Polynomial in degree of dual graph
• BT-free search: hints to problem tractability
– Various reformulations of the dual graph
– Adaptive, unifying, self-regulatory, automatic strategy
– Empirical evidence, supported by statistics
• Future work
– Extend to constraints given as conflicts, in intension
– Extend to singleton type consistencies
1/23/2012 SARA 2011 11
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