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
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

0
0
1
a b c d
1
0
0
b, d
c
a
0
1
0
1
0
0
Now the rows are indistinguishable
6
Index Symmetry in One Dimension
A B C
D E F
G H I

• 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
A B C
D E F
G H I
A B C
D E F
G H I
Consistent
Consistent
A B C
D E F
G H I
A B C
D E F
G H I
Inconsistent
Inconsistent
8
Incompleteness of Double Lex
1
0
1
0
1
0
Swap 2 columns Swap row 1 and 3
0
1
0
1
0
1
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 ? lex V2) and
• GAC(V2 ? lex V3) does not imply
• GAC(V1 ? lex V3)

• Not pair-wise decomposable

does not imply
GAC(V1 ? lex V2 ? lex ? lex Vn)
11
Complete Solution for 2x3 Matrices
C
B
A
ABCDEF is minimal among the index symmetries
F
E
D

1. ABCDEF ? ACBDFE
2. ABCDEF ? BCAEFD
3. ABCDEF ? BACEDF
4. ABCDEF ? CABFDE
5. ABCDEF ? CBAFED
6. ABCDEF ? DFEACB

1. ABCDEF ? EFDBCA
2. ABCDEF ? EDFBAC
3. ABCDEF ? FDECAB
4. ABCDEF ? FEDCBA
5. ABCDEF ? DEFABC

12
Simplifying the Inequalities
C
B
A
F
E
D

• 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
C
B
A
F
E
D
5
3
1
5
3
1
Swap 2 rows Rotate columns left
3
1
5
1
5
3
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.