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

2. Matrix Models Many CSP Problems can be modelled by a multi-dimensional matrix of decision variables. 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 

3. Matrix Models (2) Social Golfers –Set of rounds  –Set of groups  –Set of golfers  Steel Mill Slab Design Printing Template Design Warehouse Location Progressive Party Problem …

4. Transforming Value Symmetry to Index Symmetry a, b, c, d are indistinguishable values a  1000 b  0100 {a, b}  1100 {a, c}  1010 The indices of these vectors are indistinguishable.

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

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

7. Properties Incomplete in general Challenge: break all symmetries Complete 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.

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