Breaking Symmetry in Matrix Models of Constraint Satisfaction Problems

1. Breaking Symmetryin Matrix Models ofConstraint Satisfaction Problems 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

2. The Constraint Satisfaction Problem An instance of the CSP consists of • Finite set of variables X1,…,Xn, having finite domains D1,…,Dn. • 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) 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 X ! X ! X ! 0 1 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 0 1 0 X ! ! !

5. 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

6. 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)

7. 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 • …

8. 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

9. 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])

10. Index Symmetry in Multiple Dimensions Consistent Consistent Inconsistent Inconsistent

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

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 • 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)

15. 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

16. 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

17. 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)

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.