r.j. woodward 1 , s. karakashian 1 , b.y. choueiry 1 & c. bessiere 2
DESCRIPTION
Revisiting Neighborhood Inverse Consistency on Binary CSPs. R.J. Woodward 1 , S. Karakashian 1 , B.Y. Choueiry 1 & C. Bessiere 2 1 Constraint Systems Laboratory, University of Nebraska-Lincoln 2 LIRMM-CNRS, University of Montpellier. Acknowledgements - PowerPoint PPT PresentationTRANSCRIPT
Constraint Systems Laboratory
R.J. Woodward1, S. Karakashian1, B.Y. Choueiry1 & C. Bessiere2
1Constraint Systems Laboratory, University of Nebraska-Lincoln
2LIRMM-CNRS, University of Montpellier
Revisiting Neighborhood Inverse Consistency
on Binary CSPs
Acknowledgements• Experiments conducted at UNL’s Holland Computing Center• Robert Woodward supported by a NSF Graduate Research Fellowship grant number 1041000• NSF Grant No. RI-111795
04/21/23 CP 2012 1
Constraint Systems Laboratory
Outline• Introduction: Relational NIC• Structure of the dual graph of a binary CSP
affects RNIC• RNIC versus NIC, sCDC on binary CSPs• Experimental results• Conclusion
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Constraint Systems Laboratory
Constraint Satisfaction Problem
• Graphical Representation– Constraint graph– Dual graph– Minimal dual graph
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R3
CD
ACAB
R2
C
A
DAD
A A
R1
R4
Dual graph
R3
BA
C
R1
R4
R2
D
Constraint graph
R3
CD
ACAB
R2
C
A
DAD
A A
R1
R4
Minimal dual graph [Janssen+,1989]
Constraint Systems Laboratory
Neighborhood Inverse Consistency
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• Relational NIC [Woodward+ AAAI 2011]
– Reformulation of NIC [Freuder & Elfe, AAAI 1996]
– 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!)
– Initially designed for non-binary CSPs
• How about RNIC on binary CSPs?– Impact of the structure of the dual graph
– RNIC versus other consistency properties R3
R2R1
R4
A
B
C
D
Constraint Systems Laboratory
Complete Constraint Graph
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V1
V2
V3
Vn-1
Vn
C1,2
Vn Cn-1,nC1,n
V1 Vn-1
Vn C3,n C2,n
V2V3
V5 V5 V5
C2,3C1,3
C3,4C1,4 C2,4
C4,5C1,5
V1
C3,5 C2,5
V1
V1V2
V2
V2
V3 V3
V4
V3V4
V4
V5 V2V3V4V1
V1
Vn Vn C4,n
V4
Ci-2,iC1,iVi Vi
C3,i C2,i
Ci,i+1
Ci,n-1
Ci,n
Vi
Vi
Vi
Vi
(i-2) vertices
(n-i)
ver
tices
Vi
Vi
Vi
Dual Graph: Triangle shaped grids
Constraint Systems Laboratory
Minimal Dual Graph
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V1
V2
V3
Vn-1
Vn
Vn Cn-1,nC1,n
V1 Vn-1
Vn Vn C3,n C2,n
V2V3
V5 V5 V5
C1,2
C2,3C1,3
C3,4C1,4 C2,4
C4,5C1,5
V1
C3,5 C2,5
V1
V1 V2
V2
V2
V3 V3
V4
V3V4
V4
V5 V2V3V4V1
V1
Ci-2,iC1,iVi Vi
C3,i C2,i
Ci,i+1
Ci,n-1
Ci,n
Vi
Vi
Vi
Vi
(i-2) vertices
(n-i)
ver
tices
Vi
Vi
Vi
Vn C4,n
V4
Dual Graph: Triangle shaped grids
Constraint Systems Laboratory
Minimal Dual Graph
… can be a triangle-shaped grid (planar)
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V5 V5 V5
C1,2
C2,3C1,3
C3,4C1,4 C2,4
C4,5C1,5
V1
C3,5 C2,5
V1
V1V2
V2
V2
V3 V3
V4
V3V4
V4
… but does not have to be– Star on V2
– Cycle of size 6
C1,5
C1,3
C3,5
C2,4
C4,5
C3,4
V1V1
V2
V2
V3V3
V4
V4V5C1,2
C1,4
C2,3
C2,5
V1
V2V5
V5
V4
V3
V1
V2
V3 V4
C1,4 V5C2,5 C3,5
C1,2 C1,5
C2,3
C3,4
C4,5
C1,3
C2,4
Constraint Systems Laboratory
Non-Complete Constraint Graph
• Can still be a triangle-shaped grid– Have a chain of vertices– of length ≤ n-1
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V1
V2
V3 V4
C1,4
V5C2,5
C3,5
C1,2 C1,5
C2,3
C3,4
C1,2
C2,3
C3,4C1,4
C1,5
V1
C3,5 C2,5
V1
V2
V2
V3
V3V4
V5 V5
Constraint Systems Laboratory
Impact on RNIC
On a binary CSP, RNIC enforced on the minimal dual graph (wRNIC) is never strictly stronger than R(*,3)C.
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• R(*,m)C ensures that subproblem induced on the dual CSP by every connected combination of m relations is minimal
[Karakashian+, AAAI 2010]
R3
R2R1
R4
Constraint Systems Laboratory
wRNIC on Binary CSPs
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C1
C3
C2 C4
V1 V1
V2
V3
C1
C3
C2 C4
V1 V2
V1
V3
• wRNIC can never consider more than 3 relations
• In either case, it is not possible to have an edge between C3 & C4 (a common variable to C3 & C4) while keeping C3 as a binary constraint
Constraint Systems Laboratory
NIC, sCDC, and RNIC not comparable
• NIC Property [Freuder & Elfe, AAAI 1996]
↪ Every value can be extended to a solution in its variable’s neighborhood
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A
B
C
D
R3
R2R1
R4
• sCDC Property [Lecoutre+, JAIR 2011]
↪ An instantiation {(x,a),(y,b)} is DC iff (y,b) holds in SAC when x=a and (x,a) holds in SAC when y=b and (x,y) in scope of some constraint. Further, the problem is also AC.
• RNIC Property [Woodward+, AAAI 2011]
↪ Every tuple can be extended to a solution in its relation’s neighborhood
↪ wRNIC, triRNIC, wtriRNIC enforce RNIC on a minimal, triangulated, and minimal triangulated dual graph, respectively
↪ selRNIC automatically selects the RNIC variant based on the density of the dual graph
Constraint Systems Laboratory
Experimental Results (CPU Time)
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Benchmark # inst. AC3.1 sCDC1 NIC selRNICCPU Time (msec)
NIC Quickestbqwh-16-106 100/100 3,505 3,860 1,470 3,608bqwh-18-141 100/100 68,629 82,772 38,877 77,981coloring-sgb-queen 12/50 680,140 (+3) - (+9) 57,545 634,029coloring-sgb-games 3/4 41,317 33,307 (+1) 860 41,747rand-2-23 10/10 1,467,246 1,460,089 987,312 1,171,444rand-2-24 3/10 567,620 677,253 (+7) 3,456,437 677,883
sCDC Quickestdriver 2/7 (+5) 70,990 (+5) 17,070 358,790 (+4) 185,220ehi-85 87/100 (+13) 27,304 (+13) 573 513,459 (+13) 75,847ehi-90 89/100 (+11) 34,687 (+11) 605 713,045 (+11) 90,891frb35-17 10/10 41,249 38,927 179,763 73,119
RNIC Quickestcomposed-25-1-25 10/10 226 335 1,457 114composed-25-1-2 10/10 223 283 1,450 88composed-25-1-40 9/10 (+1) 288 (+1) 357 120,544 (+1) 137composed-25-1-80 10/10 223 417 (+1) - 190composed-75-1-25 10/10 2,701 1,444 363,785 305composed-75-1-2 10/10 2,349 1,733 48,249 292composed-75-1-40 7/10 (+1) 1,924 (+3) 1,647 631,040 (+3) 286composed-75-1-80 10/10 1,484 1,473 (+1) - 397
Constraint Systems Laboratory
Experimental Results (BT-free, #NV)
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Benchmark # inst. AC3.1 sCDC1 NIC selRNIC AC3.1 sCDC1 NIC selRNICBT-Free #NV
NIC Quickestbqwh-16-106 100/100 0 3 8 5 1,807 1,881 739 1,310bqwh-18-141 100/100 0 0 1 0 25,283 25,998 12,490 22,518coloring-sgb-queen 12/50 1 - 16 1 91,853 - 15,798 91,853coloring-sgb-games 3/4 1 1 4 1 14,368 14,368 40 14,368rand-2-23 10/10 0 0 10 0 471,111 471,111 12 471,111rand-2-24 3/10 0 0 10 0 222,085 222,085 24 222,085
sCDC Quickestdriver 2/7 1 2 1 1 3,893 409 3,763 3,763ehi-85 87/100 0 100 87 100 1,425 0 0 0ehi-90 89/100 0 100 89 100 1,298 0 0 0frb35-17 10/10 0 0 0 0 24,491 24,491 24,491 24,346
RNIC Quickestcomposed-25-1-25 10/10 0 10 10 10 153 0 0 0composed-25-1-2 10/10 0 10 10 10 162 0 0 0composed-25-1-40 9/10 0 10 9 10 172 0 0 0composed-25-1-80 10/10 0 10 - 10 112 0 - 0composed-75-1-25 10/10 0 10 10 10 345 0 0 0composed-75-1-2 10/10 0 10 10 10 346 0 0 0composed-75-1-40 7/10 0 10 7 10 335 0 0 0composed-75-1-80 10/10 0 10 - 10 199 0 - 0
Constraint Systems Laboratory
Conclusions
• Contributions– Apply RNIC to binary CSPs– Structure of dual graph & impact of RNIC– NIC, sCDC, and RNIC are incomparable– Empirically shown benefits of higher-level consistencies
• Future work– Study impact of the structure of the dual graph on
(future) relational consistency properties– ‘Predict’ appropriate consistency property using
information about the problem and its structure
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