Increasing Symmetry Breaking by Preserving T arget Symmetries - - PowerPoint PPT Presentation

▶
increasing symmetry breaking by preserving t arget
SMART_READER_LITE
LIVE PREVIEW

Increasing Symmetry Breaking by Preserving T arget Symmetries - - PowerPoint PPT Presentation

Nov 09, 2023 12 likes •309 views

1 Increasing Symmetry Breaking by Preserving T arget Symmetries Jimmy Lee and Jingying Li THE CHINESE UNIVERSITY OF HONG KONG 2 Overview A lot of CSPs are highly symmetric Breaking Symmetries in CSPs significantly reduce

slide-1
SLIDE 1

Increasing Symmetry Breaking by Preserving T arget Symmetries

Jimmy Lee and Jingying Li

THE CHINESE UNIVERSITY OF HONG KONG

1

slide-2
SLIDE 2

Overview

  • A lot of CSPs are highly symmetric
  • Breaking Symmetries in CSPs

significantly reduce search tree size

  • Two main methods to break symmetries

– static symmetry breaking – dynamic symmetry breaking

2

… … … … … …

slide-3
SLIDE 3

Overview

  • Generally, exponential number of symmetries →

exponential number of symmetry breaking constraints

  • Partial Symmetry Breaking

1. choose which symmetries to break [Jefferson and Petrie 2011] 2. choose what constraints to use

  • preserving target symmetries

3

Cσ1, Cσ2, Cσ3, Cσ4, Cσ5 … … Cσn-4,Cσn-3,Cσn-2,Cσn-1,Cσn

σ1, σ2, σ3, σ4, σ5, … … σn-4, σn-3, σn-2, σn-1, σn

slide-4
SLIDE 4

Symmetry

A symmetry on P (CSP Problem) for sol(P) is a bijective mapping σ on the v-vals(P) (variable value pairs) such that sol(P) σ = sol(P).

  • variable symmetry σ

(xi = ai)σ ≡ xiσ = ai

  • value symmetry σ

(xi = ai)σ ≡ xi = ai

σ

  • constraint symmetry σ

preserve the set of constraints σrx σry σd1 σd2 σr90 σr180 σr270

4

Ggeo:

slide-5
SLIDE 5

Symmetry Breaking Constraints

vertical reflection σry [x1, x2, x3, x4, x5] → σry([x1, x2, x3, x4, x5])

5 x1 x2 x3 x4 x5

slide-6
SLIDE 6

Symmetry Breaking Constraints

vertical reflection σry [x1, x2, x3, x4, x5] → [x5, x4, x3, x2, x1] LEXLEADER [Crawford et al. 1996] constraint enforces lexicographical

  • rdering on variable sequence:

X ≤lex σ(X) Cσry [x1, x2, x3, x4, x5] ≤lex [x5, x4, x3, x2, x1] [x5, x4, x3, x2, x1] ≤lex [x1, x2, x3, x4, x5]

6

→ x1 < x5 → x5 < x1 ……

slide-7
SLIDE 7

Symmetry Breaking in Search T ree

  • The goal of breaking symmetries is to avoid the

exploration of a search space with assignments that can be mapped by a representative in symmetry class via symmetry function.

7

root

X1=1 X1=n X1=2

…

slide-8
SLIDE 8

# Symmetries to eliminate Runtime

Partial Symmetry Breaking [McDonald and Smith 2002]

8

Constraint Propagation Overhead Search Time (Search Tree Size)

  • {σ1 , σ2 , σ3 , … , … , … , σn-1 , σn }

{Cσ1, Cσ2, Cσ3 , … , … , … , Cσn-1, Cσn }

Target Symmetries!

⊇ { σi1 , σi2 , …, σim } ⊇ { Cσi1 , Cσi2 , …, Cσim }

+

slide-9
SLIDE 9

# Symmetries to eliminate Runtime

Motivation and Goal

  • Select a set of symmetry breaking constraints that aims to eliminate
  • nly target symmetries but is able to eliminate as much symmetries as

possible. – similar overhead but turn out to prune more space finally

9

Constraint Propagation Overhead Search Time (Search Tree Size)

  • {σ1 , σ2 , σ3 , … , … , … , σn-1 , σn }

{Cσ1, Cσ2, Cσ3 , … , … , … , Cσn-1, Cσn }

Target Symmetries!

⊇ { σi1 , σi2 , …, σim } ⊇ { Cσi1 , Cσi2 , …, Cσim }

{ Cσi1 , Cσi2 , …, Cσim } ’ ’ ’

+

slide-10
SLIDE 10

Symmetry Preservation - Definition

Definition 4.2: Given a CSP P = (V, D, C) with a symmetry σ. The symmetry σ is preserved by a set of symmetry breaking constraints Csb iff σ is a symmetry of P’ = (V, D, C ∪ Csb).

10

none none vertical reflection horizontal reflection main diagonal reflection minor diagonal reflection

1 1 3 5 2 3 2 4 5 4 5 3 5 1 2 4 2 4 1 3 2 4 1 3 5

Diagonal Latin Square 6 choices of value symmetry breaking constraints

slide-11
SLIDE 11

Symmetry Preservation - Properties

If Cσ1 preserve σ2,

  • Theorem 4.3: Eliminating the target symmetries already

eliminates their compositions. – eliminate σ1, σ2 → eliminate σ1 ○ σ2 , σ2 ○ σ1

  • Theorem 4.4: The combination of two sound symmetry

breaking constraints is still sound. – sound Cσ1, Cσ2 → sound Cσ1 ∪ Cσ2

11

slide-12
SLIDE 12

Why is Preservation good?

  • Solution reduction of combining constraints

12

Cσ1 Cσ2

Cσ1

1

Cσ1

2

Cσ1

k

… Cσ2

1

Cσ2

2

Cσ2

k

…

+ =

half half

slide-13
SLIDE 13

Why is Preservation good?

  • guarantee good solution reduction

13

Cσ1 Cσ2

Cσ1

1

Cσ1

2

Cσ1

k

… Cσ2

1

Cσ2

2

Cσ2

k

…

+ =

preserve σ2

always ¼

slide-14
SLIDE 14

Experimental Results

  • 3GHz Intel Core2 Duo PC running Gecode-3.7.0
  • # solutions, runtime (second), # fails
  • Preserving Target Symmetries

– Diagonal Latin Square Problem – NN-Queen Problem – Cover Array Problem (ILOG6.0) – Error Correcting Code - Lee Distance

14

slide-15
SLIDE 15

Diagonal Latin Square (n)

  • ROWWISE

: [X11,X12,…,X1n] = [1,2,…,n]

  • Our Method

: [X11,X22,…,Xnn] = [1,2,…,n] , X1n < Xn1

15

n ROWWISE Our Method #sol time #fails #sol time #fails 5 8 0.001 7 4 0.001 1 6 128 0.029 3000 64 0.004 652 7 171200 12.891 1413K 85600 1.954 163K 8 0.002 140 0.001 17 9 40.04 4327K 0.001 25 10 0.031 2025 0.002 175 11 12052 1204124K 0.005 339 all solutions

__________

  • ne solution
slide-16
SLIDE 16

Diagonal Latin Square (n)

  • ROWWISE

: [X11,X12,…,X1n] = [1,2,…,n]

  • Our Method

: [X11,X22,…,Xnn] = [1,2,…,n] , X1n < Xn1

16

n ROWWISE Our Method #sol time #fails #sol time #fails 5 8 0.001 7 4 0.001 1 6 128 0.029 3000 64 0.004 652 7 171200 12.891 1413K 85600 1.954 163K 8 0.002 140 0.001 17 9 40.04 4327K 0.001 25 10 0.031 2025 0.002 175 11 12052 1204124K 0.005 339

half of the solution set size

all solutions

slide-17
SLIDE 17

Diagonal Latin Square (n)

  • ROWWISE

: [X11,X12,…,X1n] = [1,2,…,n]

  • Our Method

: [X11,X22,…,Xnn] = [1,2,…,n] , X1n < Xn1

17

n ROWWISE Our Method #sol time #fails #sol time #fails 5 8 0.001 7 4 0.001 1 6 128 0.029 3000 64 0.004 652 7 171200 12.891 1413K 85600 1.954 163K 8 0.002 140 0.001 17 9 40.04 4327K 0.001 25 10 0.031 2025 0.002 175 11 12052 1204124K 0.005 339 all solutions

slide-18
SLIDE 18

Diagonal Latin Square (n)

  • ROWWISE

: [X11,X12,…,X1n] = [1,2,…,n]

  • Our Method

: [X11,X22,…,Xnn] = [1,2,…,n] , X1n < Xn1

18

n ROWWISE Our Method #sol time #fails #sol time #fails 5 8 0.001 7 4 0.001 1 6 128 0.029 3000 64 0.004 652 7 171200 12.891 1413K 85600 1.954 163K 8 0.002 140 0.001 17 9 40.04 4327K 0.001 25 10 0.031 2025 0.002 175 11 12052 1204124K 0.005 339

  • ne solution
slide-19
SLIDE 19

Cover Array Problem [Hnich et al. 2006]

  • Target Symmetries:

– row, column {Grow , Gcol}

  • dLex [Flener et al. 2002]
  • mLex

– row, column, value {Grow, Gcol , Gval}

  • dLex-V [Law and Lee 2004]
  • mLex-V
  • Experiments are conducted with various problem

sizes CA(t,k,g,b) .

19

: CGcol [Frisch et al. 2003] preserves Grow : CGval preserves {Grow, Gcol}, CGcol preserves Grow

slide-20
SLIDE 20

Preserving T arget Symmetries - CA (log10#solutions)

20

slide-21
SLIDE 21

Cover Array Problem (log10#solutions)

21

dLex

Target Symmetries Tradition Our Method row, column dLex mLex row, column, value dLex-V mLex-V

mLex

slide-22
SLIDE 22

Cover Array Problem (log10#solutions)

22

mLex-V

Target Symmetries Tradition Our Method row, column dLex mLex row, column, value dLex-V mLex-V

dLex-V

slide-23
SLIDE 23

Cover Array Problem (log10#solutions)

23

dLex dLex-V

Target Symmetries Tradition Our Method row, column dLex mLex row, column, value dLex-V mLex-V

slide-24
SLIDE 24

Cover Array Problem (log10#solutions)

24

dLex dLex-V mLex mLex-V

Target Symmetries Tradition Our Method row, column dLex mLex row, column, value dLex-V mLex-V

slide-25
SLIDE 25

Cover Array Problem (log10#solutions)

25

dLex-V mLex

Target Symmetries Tradition Our Method row, column dLex mLex row, column, value dLex-V mLex-V

slide-26
SLIDE 26

26

Preserving T arget Symmetries - CA (runtime)

slide-27
SLIDE 27

Concluding Remarks

  • 1. formally define symmetry preservation
  • 2. propose method to post symmetry breaking

constraints for only target symmetries but they turn out to eliminate more symmetries

  • 3. demonstrate the feasibility and efficiency of our

proposals by experimental results

27

slide-28
SLIDE 28

Future Work

  • Combination with other approaches

– choose better target symmetries [Jefferson and Petrie 2011] – model restart [Heller et al. 2008] – symmetries of sb constraints [Katsirelos and Walsh 2010]

  • Automating the procedure of selecting constraints

28

slide-29
SLIDE 29

TH THANK NK YOU OU!

29