1. Revisiting Neighborhood Inverse Consistency on Binary CSPs
R.J. Woodward1, S. Karakashian1, B.Y. Choueiry1 & C. Bessiere2
1Constraint Systems Laboratory, University of Nebraska-Lincoln
2LIRMM-CNRS, University of Montpellier
Acknowledgements
Experiments conducted at UNL’s Holland Computing Center
Robert Woodward supported by a NSF Graduate Research Fellowship grant number
NSF Grant No. RI
6/6/2019
CP 2012

2. 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
Insist on Binary
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CP 2012

3. Constraint Satisfaction Problem
Graphical Representation
Constraint graph
Dual graph
Minimal dual graph R4 A B R3 R1 D C R2
Constraint graph R3 CD AC AB R2 C A D AD R1 R4
Dual graph R1 R2 D AD CD A A C AB AC A
an edge between two vertices is redundant if there exists an alternate path between the two vertices such that the shared variables appear in every vertex in the path R4 R3
Minimal dual graph
[Janssen+,1989]
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CP 2012

4. Neighborhood Inverse Consistency
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 A C B D R3 R2 R1 R4
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CP 2012

5. Complete Constraint GraphV1 V2 V3
Vn-1 Vn
Dual Graph: Triangle shaped grids
C1,2 Vn
Cn-1,n
C1,n V1
Vn-1
C3,n
C2,n V2 V3 V5
C2,3
C1,3
C3,4
C1,4
C2,4
C4,5
C1,5
C3,5
C2,5 V4
C4,n
Ci-2,i
C1,i Vi
C3,i
C2,i
Ci,i+1
Ci,n-1
Ci,n
(i-2) vertices
(n-i) vertices
Triangle shaped grids
n-1 vertices on the diagonal correspond to the constraints over V1
n-1 vertices corresponding to the constraints over Vi≥2 are located:
Centered on C1,i
i-2 vertices are lined up along the horizontal axis
n-i vertices are lined up along the vertical axis
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CP 2012

6. Dual Graph: Triangle shaped grids
Minimal Dual Graph V1 V2 V3
Vn-1 Vn
Dual Graph: Triangle shaped grids Vn Vn Vn Vn
C1,n
Cn-1,n
C4,n
C3,n
C2,n V1
Vn-1 V4 V3 V2
Ci,n V1 V5 V4 V3 V2 V5 V5 V5 Vi
C1,5
C4,5
C3,5
C2,5
Ci,n-1
(n-i) vertices V1 V4 V3 V2 Vi V4 V4 Vi Vi
C1,4
C3,4
C2,4
Ci,i+1 V1 V3 V2 V1 V3 Vi
C1,3
C2,3
C1,i
Ci-2,i
C3,i
C2,i Vi Vi Vi V1 V2
C1,2 Vi
(i-2) vertices
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CP 2012

7. Minimal Dual Graph … can be a triangle-shaped grid (planar)V1 V2 V3 V4
C1,4 V5
C2,5
C3,5
C1,2
C1,5
C2,3
C3,4
C4,5
C1,3
C2,4
… can be a triangle-shaped grid (planar)
… but does not have to be
Star on V2
Cycle of size 6 V5
C1,2
C2,3
C1,3
C3,4
C1,4
C2,4
C4,5
C1,5 V1
C3,5
C2,5 V2 V3 V4
C1,5
C1,3
C3,5
C2,4
C4,5
C3,4 V1 V2 V3 V4 V5
C1,2
C1,4
C2,3
C2,5
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CP 2012

8. Non-Complete Constraint Graph
Can still be a triangle-shaped grid
Have a chain of vertices
of length ≤ n-1
C1,2
C2,3
C3,4
C1,4
C1,5 V1
C3,5
C2,5 V2 V3 V4 V5 V1 V2 V3 V4
C1,4 V5
C2,5
C3,5
C1,2
C1,5
C2,3
C3,4
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CP 2012

9. Impact on RNIC
On a binary CSP, RNIC enforced on the minimal dual graph (wRNIC) is never strictly stronger than R(*,3)C. R3 R2 R1 R4
R(*,m)C ensures that subproblem induced on the dual CSP by every connected combination of m relations is minimal [Karakashian+, AAAI 2010]
Because of the structure of the dual graph of a binary CSP, we have:
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CP 2012

10. wRNIC on Binary CSPs 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
wRNIC, which is RNIC on the minimal dual graph
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CP 2012

11. 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 A B C D
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 R3 R2 R1 R4
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CP 2012

12. Experimental Results (CPU Time)
Benchmark
# inst.
AC3.1
sCDC1
NIC
selRNIC
CPU Time (msec)
NIC Quickest
bqwh
100/100
3,505
3,860
1,470
3,608
bqwh
68,629
82,772
38,877
77,981
coloring-sgb-queen
12/50
680,140
(+3) -
(+9) 57,545
634,029
coloring-sgb-games
3/4
41,317
33,307
(+1) 860
41,747
rand-2-23
10/10
1,467,246
1,460,089
987,312
1,171,444
rand-2-24
3/10
567,620
677,253
(+7) 3,456,437
677,883
sCDC Quickest
driver
2/7
(+5) 70,990
(+5) 17,070
358,790
(+4) 185,220
ehi-85
87/100
(+13) 27,304
(+13) 573
513,459
(+13) 75,847
ehi-90
89/100
(+11) 34,687
(+11) 605
713,045
(+11) 90,891
frb35-17
41,249
38,927
179,763
73,119
RNIC Quickest
composed
226
335
1,457
114
composed
223
283
1,450 88
composed
9/10
(+1) 288
(+1) 357
120,544
(+1) 137
composed
417
(+1) -
190
composed
2,701
1,444
363,785
305
composed
2,349
1,733
48,249
292
composed
7/10
(+1) 1,924
(+3) 1,647
631,040
(+3) 286
composed
1,484
1,473
397
NIC: Not effective on sparse problems, too costly on dense problems [Debruyene+, 01]
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CP 2012

13. Experimental Results (BT-free, #NV)
Benchmark
# inst.
AC3.1
sCDC1
NIC
selRNIC
BT-Free
#NV
NIC Quickest
bqwh
100/100 3 8 5
1,807
1,881
739
1,310
bqwh 1
25,283
25,998
12,490
22,518
coloring-sgb-queen
12/50 - 16
91,853
15,798
coloring-sgb-games
3/4 4
14,368 40
rand-2-23
10/10 10
471,111 12
rand-2-24
3/10
222,085 24
sCDC Quickest
driver
2/7 2
3,893
409
3,763
ehi-85
87/100
100 87
1,425
ehi-90
89/100 89
1,298
frb35-17
24,491
24,346
RNIC Quickest
composed
153
composed
162
composed
9/10 9
172
composed
112
composed
345
composed
346
composed
7/10 7
335
composed
199
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CP 2012

14. Conclusions Contributions Future work 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
6/6/2019
CP 2012