Alan M. Frisch

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

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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. xyltz.

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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

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Partial Instantiation Search(Forward Checking)

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Index Symmetry in Matrix Models

• Many CSP Problems can be modelled by a
  multi-dimensional matrix of decision variables.

Round Robin Tournament Schedule
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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)

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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

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Transforming Value Symmetry to Index Symmetry

• a, b, c, d are indistinguishable values

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Now the rows are indistinguishable
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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)

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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
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Incompleteness of Double Lex
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Swap 2 columns Swap row 1 and 3
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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

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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.

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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)
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Complete Solution for 2x3 Matrices
C
B
A
ABCDEF is minimal among the index symmetries
F
E
D

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

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

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Simplifying the Inequalities
C
B
A
F
E
D
1st row ? all permutations of 2nd 6. ABC ? DFE 8.
ABC ? EDF 10. ABC ? FED 11. ABC ? DEF 9. ABC ?
FDE 7. ABCD ? EFDB

• Columns are lex ordered
• 1. BE ? CF
• 3. AD ? BE

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Illustration
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Swap 2 rows Rotate columns left
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Both satisfy 7. ABC ? EFD Right one satisfies
7. ABCD ? EFDB (1353 ? 5133) Left one violates 7.
ABCD ? EFDB (1355 ? 1353)
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• Symmetry-Breaking Predicates for Search Problems
• J. Crawford, M. Ginsberg, E. Luks, A. Roy, KR
  97.

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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.