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

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 

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 …

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.

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])

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

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.

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