Breaking Symmetry in Matrix Models of Constraint Satisfaction Problems

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Breaking Symmetry in Matrix Models of Constraint 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

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### 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

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. x+y<z.
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
Partial Instantiation Search(Forward Checking)

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Week 1

Week 2

Week 3

Week 4

Week 5

Week 6

Week 7

Period 1

0 vs 1

0 vs 2

4 vs 7

3 vs 6

3 vs 7

1 vs 5

2 vs 4

Period 2

2 vs 3

1 vs 7

0 vs 3

5 vs 7

1 vs 4

0 vs 6

5 vs 6

Period 3

4 vs 5

3 vs 5

1 vs 6

0 vs 4

2 vs 6

2 vs 7

0 vs 7

Period 4

6 vs 7

4 vs 6

2 vs 5

1 vs 2

0 vs 5

3 vs 4

1 vs 3

Index Symmetry in Matrix Models
• Many CSP Problems can be modelled by a multi-dimensional matrix of decision variables.

Round Robin Tournament Schedule

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)
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
Transforming Value Symmetry to Index Symmetry
• a, b, c, d are indistinguishable values

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{b, d}

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Now the rows are indistinguishable

Index Symmetry in One Dimension
• 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])
Index Symmetry in Multiple Dimensions

Consistent

Consistent

Inconsistent

Inconsistent

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
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.
Enforcing Lexicographic Ordering
• Not transitive

GAC(V1lexV2) and

GAC(V2lexV3) does not imply

GAC(V1lexV3)

• Not pair-wise decomposable

does not imply

GAC(V1lexV2 lex … lex Vn)

Complete Solution for 2x3 Matrices

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ABCDEF is minimal among the index symmetries

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• ABCDEF  ACBDFE
• ABCDEF  BCAEFD
• ABCDEF  BACEDF
• ABCDEF  CABFDE
• ABCDEF  CBAFED
• ABCDEF  DFEACB
• ABCDEF  EFDBCA
• ABCDEF  EDFBAC
• ABCDEF  FDECAB
• ABCDEF  FEDCBA
• ABCDEF  DEFABC
Simplifying the Inequalities

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Columns are lex ordered

1. BE  CF

1st row  all permutations of 2nd

6. ABC  DFE

8. ABC  EDF

10. ABC  FED

11. ABC  DEF

9. ABC  FDE

7. ABCD EFDB

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)

Symmetry-Breaking Predicates for Search Problems

[J. Crawford, M. Ginsberg, E. Luks, A. Roy, KR ~97].

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.