Breaking symmetry in matrix models of constraint satisfaction problems
This presentation is the property of its rightful owner.
Sponsored Links
1 / 19

Breaking Symmetry in Matrix Models of Constraint Satisfaction Problems PowerPoint PPT Presentation


  • 47 Views
  • Uploaded on
  • Presentation posted in: General

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

Download Presentation

Breaking Symmetry in Matrix Models of Constraint Satisfaction Problems

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Breaking symmetry in matrix models of constraint satisfaction problems

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

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

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

Partial Instantiation Search(Forward Checking)

1

0

0

0

0

1

0

0

0

1

0

0

0

1

0

1

0

1

0

1

1

0

0

0

0

0

1

0

1

0

1

0

0

0

0

0

1

0

1

0

0

1

1

0

1

0

1

1

0

0

0

1

1

0

1

0

1

1

1

0

1

1

X

!

X

!

X

!

0

1

0

0

1

1

0

1

0

0

1

1

0

0

0

0

1

0

1

1

0

1

0

X

!

!

!


Index symmetry in matrix models

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

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 symmetry1

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

Transforming Value Symmetry to Index Symmetry

  • a, b, c, d are indistinguishable values

a

b

c

d

1

0

0

0

0

1

a

c

{b, d}

0

1

0

0

0

1

Now the rows are indistinguishable


Index symmetry in one dimension

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

Index Symmetry in Multiple Dimensions

Consistent

Consistent

Inconsistent

Inconsistent


Incompleteness of double lex

Incompleteness of Double Lex

0

1

0

1

0

1

Swap 2 columns

Swap row 1 and 3

1

0

1

0

1

0


Completeness in special cases

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

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 ordering1

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

Complete Solution for 2x3 Matrices

A

B

C

ABCDEF is minimal among the index symmetries

D

E

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

Simplifying the Inequalities

A

B

C

D

E

F

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

Illustration

A

B

C

D

E

F

1

3

5

1

3

5

Swap 2 rows

Rotate columns left

5

1

3

3

5

1

Both satisfy 7. ABC  EFD

Right one satisfies 7. ABCD EFDB(1353 5133)

Left one violates 7. ABCD EFDB(1355 1353)


Breaking symmetry in matrix models of constraint satisfaction problems

  • Symmetry-Breaking Predicates for Search Problems

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


Conclusion

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.


  • Login