1. 853 instances from the 2008 CP Solver Competition. Real full lookahead cl+proj-wR(*,m)C, which enforces R(*, m)C on each cluster, adds projection of constraints to cluster separators to bolster propagation, and uses the minimal dual graph to reduce the number of combinations. m = 2,3,4, |(cl)| (i.e., minimal clusters) 2 hours and 8 GB per instance, 853 total instances. Improving Relational Consistency Algorithms Using Dynamic Relation Partitioning A.Schneider 1, R.J.Woodward 1,2, B.Y.Choueiry 1, and C.Bessiere 2 1 Constraint Systems Laboratory University of Nebraska-Lincoln USA 2 LIRMM-CNRS University of Montpellier France Contributions From P ER T UPLE To P ER FB Experiments were conducted on the equipment of the Holland Computing Center at the University of Nebraska-Lincoln. This research was supported by NSF Grant No. RI-111795 and EU project ICON (FP7-284715). Woodward was supported by an NSF GRF Grant No. 1041000 and a Chateaubriand Fellowship. Empirical Evaluation Paper: CP 2014. September 4 t h, 2014 Relational Consistency R( ∗,m)C, m-wise consistency, ensures that every combination of m relations is minimal. Partitions: Coarse, Fine, Intermediate Future Research Extend our approach to A LL S OL, our other algorithm for enforcing minimality of m relations [Karakashian PhD 13] 1.Designed P ER FB, an algorithm for enforcing R(*,m)C, exploiting the fact that constraints in dual CSP are piecewise functional. 2.Compared performance of P ER FB and P ER T UPLE (previous algorithm) to empirically establish improvements. Piecewise Functional Constraints [Lecoutre+ ] Cumulative Charts Summary Results CPU Time (s) Number of instances completed Experimental Setup CPU Time (s) Number of instances completed For m=2,3,4, |(cl)|, P ER FB dominates P ER T UPLE m=3 performs the best, m=|(cl)| trails closely behind [Karakashian+ AAAI 13] ABE fb 6 000 fb 7 001 fb 8 010 fb 9 011 fb 10 100 fb 11 101 CF fb 20 00 fb 21 01 fb 22 10 R5R5 R2R2 ABCDG fb 1 t1t1 00000 t2t2 00010 fb 2 t3t3 00100 fb 3 t4t4 00111 fb 4 t5t5 01101 t5t5 01111 fb 5 t7t7 11111 R1R1 Samaras & Stergiou [JAIR 05] noted that the constraints in the dual CSP are piecewise functional 1.Each relation can be partitioned into blocks of equivalent tuples 2.Each block is supported by at most one other block They used above property to design PW-AC algorithm (m=2) t1t1 titi t2t2 t3t3 P ER T UPLE enforces R(*,m)C [Karakshian+ AAAI10] Given all combinations of m relations For each relation in each combination - S EARCH S UPPORT (a backtrack search with FC) ensures each tuple can be extended to the other m-1 relations - If no solution is found, tuple is removed The “subscope equality constraint” {A,B} between R 1 and R 2 determines the partition of R 1. What partition do two subscopes (e.g., {A,B}, {C}) induce on R 1 ? What partition do all the subscopes with R 1 ’s neighbors (i.e., R 2, R 3, R 4, and R 5 ) induce on R 1 ? How do those various partitions relate? How to exploit them in P ER T UPLE ? R2R2 R1R1 A,B,C,D,G A,B,E A,B,FC,FB,E,G A,B R3R3 R4R4 R5R5 We compute and store fine and coarse blocks at preprocessing. o1o1 cb 1 t1t1 t2t2 t3t3 t4t4 cb 2 t5t5 t6t6 cb 3 t7t7 o2o2 cb 4 t1t1 t2t2 t3t3 cb 5 t4t4 cb 6 t5t5 t6t6 t7t7 o3o3 cb 7 t1t1 t2t2 cb 8 t3t3 t4t4 t5t5 t6t6 t7t7 o1∪o2∪o3o1∪o2∪o3 fb 1 t1t1 t2t2 fb 2 t3t3 fb 3 t4t4 fb 4 t5t5 t6t6 fb 5 t7t7 o1∪o3o1∪o3 ib 1 t1t1 t2t2 ib 2 t3t3 t4t4 ib 3 t5t5 t6t6 ib 4 t7t7 C1C1 o1o1 C3C3 C5C5 C2C2 o2o2 o1o1 C4C4 o3o3 R1R1 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 R 1 : coarse blocks R 1 : fine R 1 : intermediate The considered set of subscopes determines the partition of R 1. ABCDG t1t1 00000 t2t2 00010 t3t3 00100 t4t4 00111 t5t5 01101 t6t6 01111 t7t7 11111 ABE 000 001 010 011 100 101 ✗ ✗ R1R1 R2R2 ∀ m-1 relations..… ∀ tuple ∀ relation m = 2m = 3m = 4m = |(cl)| Total: 853 instances P ER T UPLE P ER FB P ER T UPLE P ER FB P ER T UPLE P ER FB P ER T UPLE P ER FB #Completed546557604616566589597615 … only by516113225826 … by both541603564589 Avg. CPU (sec) 538 227 521 362 472 314 669 458 SearchSupport Calls86.40.088.126.152.719.624.78.1 ratio --3.372.693.06 〈 R 1,fb 1 〉 has support 〈 R 2,fb 6 〉, 〈 R 5,fb 20 〉. 〈 R 1,fb 2 〉 has support 〈 R 2,fb 6 〉, 〈 R 5,fb 22 〉. fb 2, fb 3 have the same signature (intermediate block {fb 2,fb 3 }). S EARCH S UPPORT is not called on fb 3.