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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
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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 (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)
Matrix Models (2) • 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 • …
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 • 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 Consistent Consistent Inconsistent 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 • BUT • Not transitive • GAC(V1lexV2) and • GAC(V2lexV3) does not imply • GAC(V1lexV3) • GAC(V1lexV2) provided by Eclipse. • Not pair-wise decomposable does not imply GAC(V1lexV2 lex … lex Vn)