Solving Difficult CSPs with Relational Neighborhood Inverse Consistency

1. Solving Difficult CSPs with Relational Neighborhood Inverse Consistency R.J. Woodward1, S. Karakashian1, B.Y. Choueiry1 & C. Bessiere2 1Constraint Systems Laboratory, University of Nebraska-Lincoln 2LIRMM-CNRS, University of Montpellier • 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 AAAI 2011

2. Outline • Introduction • Relational Neighborhood Inverse Consistency • Property, Algorithm, Improvements • Reformulating the Dual Graph by • Redundancy removal ⟿ property wRNIC • Triangulation ⟿ property triRNIC • Selection strategy: four alternative dual graphs • Experimental Results • Conclusion AAAI 2011

3. Constraint Satisfaction Problem R6 B • Warning • Consistency properties vs. algorithms • CSP • Variables, Domains • Constraints: Relations & scopes • Representation • Hypergraph • Dual graph • Solved with • Search • Enforcing consistency A Hypergraph R4 E R1 R2 R5 R3 C F D R5 R3 R1 C D AD BCD CF Dual graph A B BD AD F AB ABDE EF AB E R6 R4 R2 AAAI 2011

4. Neighborhood Inverse Consistency • Binary CSPs [Debruyene+ 01] • Not effective on sparse problems • Too costly on dense problems • Non-binary CSPs? • Neighborhoods likely too large • 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 R4 A C 0,1,2 0,1,2 R0 R1 R3 B D 0,1,2 0,1,2 R2 R6 B A R4 E R1 R2 R5 C F D R3 AAAI 2011

5. Relational NIC B A R4 E R1 R2 R5 R3 C F D Hypergraph R5 R3 R1 C D AD BCD CF A B BD AD F • Domain filtering • Property: RNIC+DF • Algorithm: Projection AB ABDE EF AB E R6 R4 R2 Dual graph • 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 AAAI 2011

6. Characterizing RNIC • GAC, SGAC • Variable-based properties • So far, most popular for non-binary CSPs R(*,m)C • Relation-based property • Every tuple has a support in every subproblem induced by a combination of m connected relations RNIC+DF R(*,2)C+DF GAC SGAC p p’ : pis strictly weaker than p’ AAAI 2011 R(*,2)C R(*,3)C RNIC R(*,δ+1)C R5 R3 R1 R6 R4 R2

7. Algorithm for Enforcing RNIC • Two queues • Q: relations to be updated • Qt(R): The tuples of relation Rwhose supports must be verified • SearchSupport(τ,R) • Backtrack search on Neigh(R) • Loop Q until is empty τi ✗ τ √ ..… • Complexity • Space: O(ketδ) • Time: O(tδ+1eδ) • Efficient for a fixed δ R Neigh(R) AAAI 2011

8. Improving Algorithm’s Performance • UseIndexTree[Karakashian+ AAAI10] • To quickly check consistency of 2 tuples • Dynamically detect dangles • Tree structures may show in subproblem @ each instantiation • Apply directional arc consistency 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 R5 R3 R1 R6 R4 R2 AAAI 2011

9. Reformulating the Dual Graph • High degree • Large neighborhoods • High computational cost • Redundancy Removal (wRNIC) • Use minimal dual graph RR+Triangulation (wtriRNIC) • Local, complementary, do not ‘clash’ • Cycles of length ≥ 4 • Hampers propagation • RNICR(*,3)C • Triangulation (triRNIC) • Triangulate dual graph R5 R3 R1 C D AD BCD CF A B BD F AD R5 R3 R1 AB ABDE EF AB E C D AD BCD CF R6 R4 R2 A B BD F D A AB ABDE EF B A E R6 R4 R2 RNIC wRNIC triRNIC wtriRNIC AAAI 2011

10. 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 Start No Yes dGo≥ 15% No No Yes Yes dGtri≤ 2 dGo dGwtri≤ 2 dGw Gwtri Gtri Gw Go AAAI 2011

11. 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 • EquivCPU: Equivalence classes based on CPU • EquivCmp: Equivalence classes based on completion • #C: Number of instances completed • #BT-free: # instances solved backtrack free AAAI 2011

12. 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 wR(*,4)C R(*,4)C R(*,2)C≡ wR(*,2)C wR(*,3)C R(*,3)C RNIC R(*,δ+1)C wR(*,δ+1)C wRNIC triRNIC wtriRNIC AAAI 2011

13. Check Poster inStudent Poster Session Wednesday, Aug10 @ 6:30PM Grand Ballroom AAAI 2011