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R.J. Woodward 1 , S. Karakashian 1 , B.Y. Choueiry 1 & C. Bessiere 2

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

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R.J. Woodward 1 , S. Karakashian 1 , B.Y. Choueiry 1 & C. Bessiere 2

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  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 1041000 • NSF Grant No. RI-111795 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 CP 2012

  3. Constraint Satisfaction Problem R4 • Graphical Representation • Constraint graph • Dual graph • Minimal dual graph A B R3 R1 D C R2 Constraint graph R1 R2 D R1 R2 AD CD D AD CD A C A A C A AB AC A AB AC R4 R3 A R4 R3 Minimal dual graph [Janssen+,1989] Dual graph CP 2012

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

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

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

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

  8. Non-Complete Constraint Graph V5 V5 C1,5 C3,5 C2,5 • Can still be a triangle-shaped grid • Have a chain of vertices • of length ≤ n-1 V1 V3 V1 V4 C1,2 C1,5 C1,4 C1,4 C3,4 V2 C2,5 V2 V5 V3 C3,5 C2,3 V1 C2,3 V2 V3 V4 C3,4 C1,2 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. R1 R2 R4 R3 • R(*,m)C ensures that subproblem induced on the dual CSP by every connected combination of m relations is minimal [Karakashian+, AAAI 2010] CP 2012

  10. wRNIC on Binary CSPs • wRNIC can never consider more than 3 relations C1 C1 V1 V2 V1 V1 C2 C4 V1 C2 C4 V2 • 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 V3 V3 C3 C3 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 • 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 R1 R2 R4 R3 CP 2012 A C B D

  12. Experimental Results (CPU Time) CP 2012

  13. Experimental Results (BT-free, #NV) CP 2012

  14. 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 CP 2012

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