1. Constraint Systems Laboratory 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 Solving Difficult CSPs with Relational Neighborhood Inverse Consistency 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 12/13/2015AAAI 20111

2. Constraint Systems Laboratory Outline Introduction Relational Neighborhood Inverse Consistency –Property, Algorithm, Improvements Reformulating the Dual Graph by 1.Redundancy removal property wRNIC 2.Triangulation property triRNIC Selection strategy: four alternative dual graphs Experimental Results Conclusion 12/13/2015AAAI 20112

3. Constraint Systems Laboratory Constraint Satisfaction Problem CSP –Variables, Domains –Constraints: Relations & scopes Representation –Hypergraph –Dual graph Solved with –Search –Enforcing consistency 12/13/2015AAAI 20113 R4R4 BCD ABDE CF EF AB R3R3 R1R1 R2R2 C F E BD AB D AD A B R5R5 R6R6 R3R3 A B C D E F R1R1 R4R4 R2R2 R5R5 R6R6 Hypergraph Dual graph Warning –Consistency properties vs. algorithms

4. Constraint Systems Laboratory Neighborhood Inverse Consistency Property [Freuder+ 96] ↪ Every value can be extended to a solution in its variable’s neighborhood ↪ Domain-based property Algorithm  No space overhead  Adapts to graph connectivity 12/13/2015AAAI 20114 0,1,2 R0R0 R1R1 R3R3 R2R2 R4R4 A B C D R3R3 A B C D E F R1R1 R4R4 R2R2 R5R5 R6R6 Binary CSPs [Debruyene+ 01]  Not effective on sparse problems  Too costly on dense problems Non-binary CSPs?  Neighborhoods likely too large

5. Constraint Systems Laboratory Relational NIC 12/13/2015AAAI 20115 Property ↪ Every tuple can be extended to a solution in its relation’s neighborhood ↪ Relation-based property Algorithm –Operates on dual graph –Filters relations –Does not alter topology of graphs R4R4 BCD ABDE CF EF AB R3R3 R1R1 R2R2 C F E BD AB D AD A B R5R5 R6R6 R3R3 A B C D E F R1R1 R4R4 R2R2 R5R5 Hypergraph Dual graph Domain filtering –Property: RNIC+DF –Algorithm: Projection

6. Constraint Systems Laboratory GAC, SGAC Variable-based properties So far, most popular for non- binary CSPs Characterizing RNIC R(*,m)C Relation-based property Every tuple has a support in every subproblem induced by a combination of m connected relations 12/13/2015AAAI 20116 GAC R(*,2)C+DF SGAC R(*,3)CR(*,δ+1)CR(*,2)C p’p : p is strictly weaker than p’ R4R4 R3R3 R1R1 R2R2 R5R5 R6R6

7. Constraint Systems Laboratory Algorithm for Enforcing RNIC Two queues 1.Q : relations to be updated 2.Q t (R) : The tuples of relation R whose supports must be verified S EARCH S UPPORT ( τ, R ) –Backtrack search on Neigh( R ) Loop Q until is empty 12/13/2015AAAI 20117 Complexity –Space: O ( ketδ ) –Time: O ( t δ+1 eδ ) –Efficient for a fixed δ..… Neigh( R ) R τ τiτi ✗ √

8. Constraint Systems Laboratory Improving Algorithm’s Performance 1.Use IndexTree [Karakashian+ AAAI10] –To quickly check consistency of 2 tuples 2.Dynamically detect dangles –Tree structures may show in subproblem @ each instantiation –Apply directional arc consistency 12/13/2015AAAI 20118 R4R4 R3R3 R1R1 R2R2 R5R5 R6R6 Note that exploiting dangles is –Not useful in R(*,m)C: small value of m, subproblem size –Not applicable to GAC: does not operate on dual graph

9. Constraint Systems Laboratory Reformulating the Dual Graph 12/13/2015AAAI 20119 R4R4 BCD ABDE CF EFAB R3R3 R1R1 R2R2 C F E BD A D AD A A B R5R5 R6R6 Cycles of length ≥ 4 –Hampers propagation –RNIC  R(*,3)C Triangulation (triRNIC) –Triangulate dual graph RR+Triangulation (wtriRNIC) Local, complementary, do not ‘clash’ wRNIC RNIC wtriRNIC triRNIC R4R4 BCD ABDE CF EFAB R3R3 R1R1 R2R2 C F E BD AB D AD A B R5R5 R6R6 High degree –Large neighborhoods –High computational cost Redundancy Removal (wRNIC) –Use minimal dual graph D B

10. Constraint Systems Laboratory Selection Strategy: Which? When? Density of dual graph ≥ 15% is too dense –Remove redundant edges Triangulation increases density no more than two fold –Reformulate by triangulation Each reformulation executed at most once 12/13/2015AAAI 201110 No YesNo Yes No d G o ≥ 15% d G tri ≤ 2 d G o d Gw tri ≤ 2 d G w GoGo G wtri GwGw G tri Start

11. Constraint Systems Laboratory Experimental Results Statistical analysis on CP benchmarks Time: Censored data calculated mean Rank: Censored data rank based on probability of survival data analysis #F: Number of instances fastest 12/13/2015AAAI 201111 AlgorithmTimeRank#FEquivCPU#CEquivCmp#BT-free 169 instances: aim-100,aim-200,lexVg,modifiedRenault,ssa wR(*,2)C944924352A138B79 wR(*,3)C92500448B134B92 wR(*,4)C116126152B132B108 GAC1711511783C119C33 RNIC6161391819C100C66 triRNIC301716999C84C80 wRNIC118484468B131B84 wtriRNIC93790423B144B129 selRNIC751586117A159A142 EquivCPU: Equivalence classes based on CPU EquivCmp: Equivalence classes based on completion #C: Number of instances completed #BT-free: # instances solved backtrack free

12. Constraint Systems Laboratory Conclusions Contributions –RNIC –Algorithm Polynomial for fixed-degree dual graphs 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 singleton-type consistencies 12/13/2015AAAI 201112 R(*,3)C wRNIC R(*,4)C RNIC wtriRNIC triRNIC R(*,δ+1)C wR(*,3)C wR(*,4)C wR(*,δ+1)C R(*,2)C ≡ wR(*,2)C

13. Constraint Systems Laboratory 12/13/2015AAAI 201113 Check Poster in Student Poster Session Wednesday, Aug10 @ 6:30PM Grand Ballroom Check Poster in Student Poster Session Wednesday, Aug10 @ 6:30PM Grand Ballroom