1. Alan M. Frisch Artificial Intelligence Group Department of Computer Science University of York Co-authors Ian Miguel, Toby Walsh, Pierre Flener, Brahim Hnich, Zeynep Kiziltan, Justin Pearson Acknowledgement Warwick Harvey Breaking Symmetry in Matrix Models of Constraint Satisfaction Problems

2. The Constraint Satisfaction Problem An instance of the CSP consists of Finite set of variables X 1,…,X n, having finite domains D 1,…,D n. Finite set of constraints. Each restricts the values that the variables can simultaneously take. Example: x neq y. x+y<z.

3. Solutions of a CSP Instance A total instantiation maps each variable to an element in its domain. A solution to a CSP instance is a total instantiation that satisfies all the constraints. Problem: Given an instance –Determine if it is satisfiable (has a solution) –Find a solution –Find all solutions –Find optimal solution

4. Partial Instantiation Search (Forward Checking) 0 0 1 00 011 110 00 00 01 10 0 00 10 10 01 1 0 0 10 11 00 10 010 110 010 11 100 101 1 X X 10 00 010 001 110 011 0 X ! 00 01 0 X 10 010 1 !! !! !

5. Index Symmetry in Matrix Models Many CSP Problems can be modelled by a multi-dimensional matrix of decision variables. 0 vs 72 vs 72 vs 60 vs 41 vs 63 vs 54 vs 5Period 3 0 vs 5 1 vs 4 3 vs 7 Week 5 3 vs 4 0 vs 6 1 vs 5 Week 6 1 vs 31 vs 22 vs 54 vs 66 vs 7Period 4 5 vs 65 vs 70 vs 31 vs 72 vs 3Period 2 2 vs 43 vs 64 vs 70 vs 20 vs 1Period 1 Week 7Week 4Week 3Week 2Week 1 Round Robin Tournament Schedule

6. Examples of Index Symmetry Balanced Incomplete Block Design –Set of Blocks  –Set of objects in each block  Rack Configuration –Set of cards (PI) –Set of rack types –Set of occurrences of each rack type 

7. Examples of Index Symmetry Social Golfers –Set of rounds  –Set of groups  –Set of golfers  Steel Mill Slab Design Printing Template Design Warehouse Location Progressive Party Problem …

8. Transforming Value Symmetry to Index Symmetry a, b, c, d are indistinguishable values {b, d}ca 100 010 100 001abcdabcd Now the rows are indistinguishable

9. Index Symmetry in One Dimension Indistinguishable Rows ABC DEF GHI 2 Dimensions [A B C]  lex [D E F]  lex [G H I] N Dimensions flatten([A B C])  lex flatten([D E F])  lex flatten([G H I])

10. Index Symmetry in Multiple Dimensions ABC DEF GHI ABC DEF GHI ABC DEF GHI ABC DEF GHI Consistent Inconsistent

11. Incompleteness of Double Lex 01 01 10 01 10 10 Swap 2 columns Swap row 1 and 3

12. Completeness in Special Cases All variables take distinct values –Push largest value to a particular corner, and –Order the row and column containing that value 2 distinct values, one of which has at most one occurrence in each row or column. –Lex order the rows and the columns Each row is a different multiset of values –Multiset order the rows and lex order the columns

13. Enforcing Lexicographic Ordering We have developed a linear time algorithm for enforcing generalized arc-consistency on a lexicographic ordering constraint between two vectors of variables. Experiments show that in some cases it is vastly superior to previous consistency algorithms, both in time and in amount pruned.

14. Enforcing Lexicographic Ordering does not imply GAC(V 1  lex V 2  lex …  lex V n ) Not pair-wise decomposable Not transitive GAC(V 1  lex V 2 ) and GAC(V 2  lex V 3 ) does not imply GAC(V 1  lex V 3 )

15. Complete Solution for 2x3 Matrices 1.ABCDEF  ACBDFE 2.ABCDEF  BCAEFD 3.ABCDEF  BACEDF 4.ABCDEF  CABFDE 5.ABCDEF  CBAFED 6.ABCDEF  DFEACB FED CBA ABCDEF is minimal among the index symmetries 7.ABCDEF  EFDBCA 8.ABCDEF  EDFBAC 9.ABCDEF  FDECAB 10.ABCDEF  FEDCBA 11.ABCDEF  DEFABC

16. Simplifying the Inequalities Columns are lex ordered 1. BE  CF 3. AD  BE 1st row  all permutations of 2 nd 6. ABC  DFE 8. ABC  EDF 10. ABC  FED 11. ABC  DEF 9. ABC  FDE 7. ABCD  EFDB FED CBA

17. Illustration Swap 2 rows Rotate columns left Both satisfy 7. ABC  EFD Right one satisfies 7. ABCD  EFDB (1353  5133) Left one violates 7. ABCD  EFDB (1355  1353) FED CBA 315 531 153 531

18. Symmetry-Breaking Predicates for Search Problems [J. Crawford, M. Ginsberg, E. Luks, A. Roy, KR ~97].

19. Conclusion Many problems have models using a multi- dimensional matrix of decision variables in which there is index symmetry. Constraint toolkits should provide facilities to support this. We have laid some foundations towards developing such facilities. Open problems remain.