Reducing Symmetry in Matrix Models Alan Frisch, Ian Miguel, Toby Walsh (York) Pierre Flener, Brahim Hnich, Zeynep Kiziltan, Justin Pearson (Uppsala)

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

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 

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 …

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

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])

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

Incompleteness of Double Lex Swap 2 columns Swap row 1 and 3

Completeness in Special Cases All variables take distinct values –Push largest value to a particular corner 2 distinct values, one of which has at most one occurrence in each row or column.

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 )

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

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

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

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

Conclusion Many problems have models using a mult- 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.