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Reformulating Dual Graphs of CSPs to Improve RNIC Performance

This paper explores reformulating the dual graphs of CSPs to enhance the performance of Relational Neighborhood Inverse Consistency (RNIC) algorithms. The reformulation involves removing redundant edges and triangulation, resulting in two alternative dual graphs. Experimental results show improvements in solving CP benchmarks.

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Reformulating Dual Graphs of CSPs to Improve RNIC Performance

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  1. Reformulating the Dual Graphs of CSPs to Improve the Performance of RNIC R.J. Woodward, S. Karakashian, B.Y. Choueiry & C. Bessiere Constraint Systems Laboratory, University of Nebraska-Lincoln LIRMM-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 SARA 2011

  2. Outline • Introduction • Relational Neighborhood Inverse Consistency • Property & algorithm • Reformulating the Dual Graph by • Removing redundant edges, yields property wRNIC • Triangulation, yields property triRNIC • Selection strategy: four alternative dual graphs • Experimental Results • Conclusion SARA 2011

  3. Constraint Satisfaction Problem R6 B • Warning • Consistency properties vs. algorithms • CSP • Variables, Domains • Constraints: binary / non-binary • 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 SARA 2011

  4. Neighborhood Inverse Consistency [Freuder+ 96] • Non-binary CSPs? • Neighborhoods likely too large • Property • Defined for binary CSPs • Every value can be extended to a solution in its variable’s neighborhood • Algorithm • No space overhead • Adapts to the connectivity • Not effective on sparse problems • To costly on dense problems 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 SARA 2011

  5. Relational NIC [Woodward+ AAAI11] B A R4 E R1 R2 R5 R3 C F D Hypergraph R5 R3 R1 C D AD BCD CF • Domain filtering • Property: RNIC+DF • Algorithm: Projection A B BD AD F AB ABDE EF AB E R6 R4 R2 Dual graph • Property • 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!) SARA 2011

  6. Reformulation: Removing Redundant Edges • High density • Large neighborhoods • Higher cost of RNIC • Minimal dual graph • Equivalent CSP • Computed efficiently [Janssen+ 89] • Run algorithm on a minimal dual graph • Smaller neighborhoods, solution set not affected • wRNIC: a strictly weaker property R5 R3 R1 C D AD BCD CF A B BD F AD AB ABDE EF AB E R6 R4 R2 dGo= 60% dGw = 40% wRNIC RNIC SARA 2011

  7. Reformulation: Triangulation • Cycles of length ≥ 4 • Hampers propagation • Triangulating dual graph • Equivalent CSP • Min-fill heuristic • Run algorithm on a triangulated dual graph • Created loops enhance propagation • triRNIC: a strictly stronger property R5 R3 R1 C D AD BCD CF A B BD F AD AB ABDE EF AB E R6 R4 R2 dGo= 60% dGtri = 67% wRNIC RNIC triRNIC SARA 2011

  8. Reformulation: RR & Triangulation R5 R3 R1 • Fixing the dual graph • RR copes with high density • Triangulation boosts propagation • RR+Tri • Both operate locally • Are complementary, do not ‘clash’ • Run algorithm on a RR+tri dual graph • CSP solution set is not affected • wtriRNIC is not comparable with RNIC C D AD BCD CF A B BD F AD AB ABDE EF AB E R6 R4 R2 dGo= 60% R5 R3 R1 C D AD BCD CF A B BD F AD AB ABDE EF AB E R6 R4 R2 dGwtri = 47% RNIC wRNIC triRNIC wtriRNIC SARA 2011

  9. Selection Strategy: Which? When? • Density ≥ 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 Yes No Yes dGtri≤ 2 dGo dGwtri≤ 2 dGw Go Gtri Gw Gwtri SARA 2011

  10. Experimental Results • Statistical analysis on CP benchmarks • Time: Censored data calculated mean • R: Censored data rank based on probability of survival data analysis • S: Equivalence classes based on CPU • SB: Equivalence classes based on completion • #C: Number of instances completed • #F: Number of instancesfastest • #BF: # instances solved backtrack free SARA 2011

  11. Conclusions • Contributions • Algorithm • Polynomial in degree of dual graph • 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 constraints given as conflicts, in intension • Extend to singleton type consistencies SARA 2011

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