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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 Symmetry in Matrix Models of Constraint Satisfaction Problems

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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

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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])

Consistent

Consistent

Inconsistent

Inconsistent

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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.