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 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


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

A

B

C

ABCDEF is minimal among the index symmetries

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F

  • 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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B

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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


Illustration

A

B

C

D

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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.


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