Reducing Symmetry in Matrix Models

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

2. Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Period 1 0 vs 1 0 vs 2 4 vs 7 3 vs 6 3 vs 7 1 vs 5 2 vs 4 Period 2 2 vs 3 1 vs 7 0 vs 3 5 vs 7 1 vs 4 0 vs 6 5 vs 6 Period 3 4 vs 5 3 vs 5 1 vs 6 0 vs 4 2 vs 6 2 vs 7 0 vs 7 Period 4 6 vs 7 4 vs 6 2 vs 5 1 vs 2 0 vs 5 3 vs 4 1 vs 3 Index Symmetry in Matrix Models • Many CSP Problems can be modelled by a multi-dimensional matrix of decision variables. Round Robin Tournament Schedule

3. Examples of Index Symmetry • Balanced Incomplete Block Design • Set of Blocks (I) • Set of objects in each block (I) • Rack Configuration • Set of cards (PI) • Set of rack types • Set of occurrences of each rack type (I)

4. Examples of Index Symmetry • Social Golfers • Set of rounds (I) • Set of groups(I) • Set of golfers(I) • Steel Mill Slab Design • Printing Template Design • Warehouse Location • Progressive Party Problem • …

5. Transforming Value Symmetry to Index Symmetry • a, b, c, d are indistinguishable values a b c d 1 0 0 0 0 1 a c {b, d} 0 1 0 0 0 1 Now the rows are indistinguishable

6. Index Symmetry in One Dimension • Indistinguishable Rows • 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])

7. Index Symmetry in Multiple Dimensions Consistent Consistent Inconsistent Inconsistent

8. Incompleteness of Double Lex 0 1 0 1 0 1 Swap 2 columns Swap row 1 and 3 1 0 1 0 1 0

9. 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.

10. Enforcing Lexicographic Ordering • Not transitive GAC(V1lexV2) and GAC(V2lexV3) does not imply GAC(V1lexV3) • Not pair-wise decomposable does not imply GAC(V1lexV2 lex … lex Vn)

11. Complete Solution for 2x3 Matrices A B C ABCDEF is minimal among the index symmetries D E F • ABCDEF  ACBDFE • ABCDEF  BCAEFD • ABCDEF  BACEDF • ABCDEF  CABFDE • ABCDEF  CBAFED • ABCDEF  DFEACB • ABCDEF  EFDBCA • ABCDEF  EDFBAC • ABCDEF  FDECAB • ABCDEF  FEDCBA • ABCDEF  DEFABC

12. Simplifying the Inequalities A B C D E F Columns are lex ordered 1. BE  CF 3. AD  BE 1st row  all permutations of 2nd 6. ABC  DFE 8. ABC  EDF 10. ABC  FED 11. ABC  DEF 9. ABC  FDE 7. ABCD EFDB

13. Illustration A B C D E F 1 3 5 1 3 5 Swap 2 rows Rotate columns left 5 1 3 3 5 1 Both satisfy 7. ABC  EFD Right one satisfies 7. ABCD EFDB(1353 5133) Left one violates 7. ABCD EFDB(1355 1353)

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

15. 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.