Alan M. FrischArtificial 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

Breaking Symmetryin Matrix Models of

Constraint Satisfaction Problems

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)

0

0

1

00

0 1 1

1 10

0 0

00

0 1

10

0

00

1 0

10

0 1

1

0

0

10

1 1

00

1 0

0 10

1 1 0

0 10

1 1

100

1 0 1

1

X X

10

0 0

0 10

0 0 1

1 10

0 1 1

0

X !

00

0 1

0

X

10

0 1 0

1

! !

! ! !

Index Symmetry in Matrix Models

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

0 vs 72 vs 72 vs 60 vs 41 vs 63 vs 54 vs 5Period 3

0 vs 5

1 vs 4

3 vs 7

Week 5

3 vs 4

0 vs 6

1 vs 5

Week 6

1 vs 31 vs 22 vs 54 vs 66 vs 7Period 4

5 vs 65 vs 70 vs 31 vs 72 vs 3Period 2

2 vs 4 3 vs 64 vs 70 vs 20 vs 1Period 1

Week 7Week 4Week 3Week 2Week 1

Round Robin Tournament Schedule

Examples of Index Symmetry

• Balanced Incomplete Block Design– Set of Blocks – Set of objects in each block

• Rack Configuration– Set of cards (PI)– Set of rack types– Set of occurrences of each rack type

Examples of Index Symmetry

• Social Golfers– Set of rounds – Set of groups– Set of golfers

• 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

{b, d}ca

100

010

100

001a

b

c

d

Now the rows are indistinguishable

Index Symmetry in One Dimension

• Indistinguishable Rows

A B C

D E F

G H I

• 2 Dimensions[A B C] lex [D E F] lex [G H I]

• N Dimensionsflatten([A B C]) lex

flatten([D E F]) lex flatten([G H I])

Index Symmetry in Multiple Dimensions

A B C

D E F

G H I

A B C

D E F

G H I

A B C

D E F

G H I

A B C

D E F

G H IConsistent Consistent

Inconsistent Inconsistent

Incompleteness of Double Lex

01

01

10

01

10

10

Swap 2 columnsSwap row 1 and 3

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

)( jlexiji

VVGAC

does not imply

GAC(V1 lex V2 lex … lex Vn)

• Not pair-wise decomposable

• Not transitive GAC(V1 lex V2) and GAC(V2 lex V3) does not imply GAC(V1 lex V3)

Complete Solution for 2x3 Matrices

1. ABCDEF ACBDFE

2. ABCDEF BCAEFD

3. ABCDEF BACEDF

4. ABCDEF CABFDE

5. ABCDEF CBAFED

6. ABCDEF DFEACB

FED

CBAABCDEF is minimal among the index symmetries

7. ABCDEF EFDBCA

8. ABCDEF EDFBAC

9. ABCDEF FDECAB

10. ABCDEF FEDCBA

11. ABCDEF DEFABC

Simplifying the Inequalities

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

FED

CBA

Illustration

Swap 2 rowsRotate columns left

Both satisfy 7. ABC EFDRight one satisfies 7. ABCD EFDB (1353 5133)Left one violates 7. ABCD EFDB (1355 1353)

FED

CBA

315

531

153

531

• 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.