Survey of Techniques for Detecting and Exploiting Symmetry in Constraint Satisfaction Problems

11/22/2005 Zheng – Comprehensive 1

Constraint Systems Laboratory

Survey of Techniques for Detecting and Exploiting

Symmetry in Constraint Satisfaction Problems

Yaling Zheng




Outline• Definition of a CSP• Motivation of the study of symmetry in C

SP• Definitions of symmetry in a CSP• Techniques of symmetry exploitation• Techniques of symmetry detection• Summary & future research directions



Definition of a CSP [from Dr. Choueiry's slides for CSCE421] • Given P = (V, D, C )

– V is a set of variables,

– D is a set of variable domains (domain values)

– C is a set of constraints,

• Query: can we find a value for each variable such that all constraints are satisfied?

nVVV ,,, 21 V

nVVV DDD ,,, 21 D

iVbVaVVVV DDDzyxiba

),,,(,...,,C with

lCCC ,,, 21 C



5-queens problemRotate 90

QQ

QQ

Q

A solution

QQ

QQ

Q

QQ

QQ

Q

Rotate 180 Rotate 270

Q

QQ

QQ

Flip horizontally

QQ

Q

Q

Q

Flip vertically Flip \ diagonally

QQ

QQ

Q

Flip / diagonally

Q

Q

Q

Q

Q

QQ

QQ

Q



Round robin tournaments• Given:

– 8 teams over 7 weeks – 4 periods per week– 2 slots per period

• Constraints: – Every team plays each week– Every team plays at most twice in each period– Every team plays every other team

• Query: Can we find a schedule which assigns a team for every slot?



A schedule to the tournament

Week 1 Week2 Week 3 Week 4 Week 5 Week 6 Week 7Period 1 0 v 1 0 v 2 4 v 7 3 v 6 3 v 7 1 v 5 2 v 4Period 2 2 v 3 1 v 7 0 v 3 5 v 7 1 v 4 0 v 6 5 v 6Period 3 4 v 5 3 v 5 1 v 6 0 v 4 2 v 6 2 v 7 0 v 7Period 4 6 v 7 4 v 6 2 v 5 1 v 2 0 v 5 3 v 4 1 v 3

• Every permutation of the week is a solution (7!)• Every permutation of the period is a solution (4!)• Every permutation of the slot is a solution (228)• Total: (7!)(4!)(228) such permutations



Motivation of exploiting symmetry• From the viewpoint of the backtrack search

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4 1 2 3 4

1 2 3 4

1 2 3 4

V1

V2

V3

V4

……

……

a state solution 1 solution 2



Question:1. What is the formal definition of symmet

ry in a CSP?2. How do we exploit the symmetries withi

n a CSP to solve it? 3. How do we detect the symmetries withi

n a given CSP so that we can exploit them?







Definitions of symmetry• Functional interchangeability [Freuder’91] • Solution symmetry [Cohen et al.’05] • Defined as a bijective mapping between solutions

(i.e., complete assignments of values to variables): {Vi,aj|Vi V , ajDVi} {Vi,aj |ViV , aj DVi}

• Maps – solutions to solutions – non-solutions to non-solutions



Relationship of symmetry concepts

Solution SymmetryInterchangeability

Partial Interchangeability

ConstraintSymmetry

FullInterchangeability

NeighborhoodInterchangeability

ValueSymmetry

VariableSymmetry

Matrix Symmetry

Row Symmetry Column Symmetry

Row and Column Symmetry[Freuder’91, Puget’02, Flenner’02, Cohen et al.’05]



Outline• Definition of a CSP• Motivation of the study of symmetry in CSP• Definitions of symmetry in a CSP• Techniques of symmetry exploitation

– CSP remodeling, propagation, bundling, static symmetry breaking, dynamic symmetry breaking, heuristic symmetry breaking

• Techniques of symmetry detection• Summary & future research directions



Techniques of exploiting symmetries

1. Remodeling a given CSP to a model with fewer symmetries

2. Exploiting symmetries during constraint propagation

3. Using bundling techniques (group symmetrical values together)



4. Adding static symmetry breaking constraints before search

5. Dynamic symmetry breaking during search

6. Using symmetry-breaking heuristic to guide search

Techniques of exploiting symmetries



Remodeling• Remove a subset of dimensions from a multi-dimen

sional model [Freuder’97]

• Model a CSP as a set domain CSP which can be solved by Conjunto [Gervet’97]– As far as I know, Conjunto is the only existing constraint logic prog

ramming language for solving set domain CSPs• Automation of remodeling (CGRASS) by condition-a

ction rules [Frisch et al.’02]



Remodeling 4-queens prob.Original model (search space 416)Variables: 4 queens Q1, Q2, Q3, Q4 on 44 boardDomains: DQ1=DQ2=DQ3=DQ4= {1,1, …, 4,4}Constraints: no two queens in the same row/colu

mn/diagonalNew model (search space 44)Variables: 4 queens Q1, Q2, Q3, Q4 in 4 rowsDomains: DQ1=DQ2=DQ3=DQ4= {1,2,3,4}Constraints: no two queens in the same column/di

agonal



Constraint propagation • Constraint propagation is to remove values f

rom variable domains by enforcing consistency techniques

• Exploit symmetry during constraint propagation – symAC2001 [Gent and MacDonald’03] symPC2001/3.1

[Gent et al.’05]– When a value of a variable is consistent (inconsis

tent), its symmetrical values of the variable is also consistent (inconsistent).



Bundling Techniques• Group neighborhood interchangeable values

together – before search [Benson and Freuder’92, Haselbock’93]

– during search [Beckwith et al.’01, Lal and Choueiry’03]

• Technique: – discrimination tree, domain partition

• Find solution bundles• Dynamic bundling performs better than static

bundling



Symmetry-breaking constraints• Add constraints over symmetrical variables before search• Examples of such constraints

– Symmetry breaking predicates [Crawford’96]– Lexicographical ordering constraints [Flener’02]– Multiset ordering constraints [Kiziltan’04]

• Good for finding all solutions by BT search• Sometimes not good for finding 1 solution• Use some (not all) symmetry-breaking constraints

before search [Puget’03]



Dynamic symmetry breaking• Symmetry excluding search (SES)

[Backofen and Will’99,’02]

• Symmetry-breaking during search (SBDS) [G

ent and Smith’00] • Symmetry-breaking via dominance detectio

n (SBDD) [Fahle et al.’01]

• CGT+SBDS [Gent et al.’02]

• CGT+SBDD [Gent et al.’03]



SES, SBDS, short SBDSQ1 2 Q1 2

Q2 = 1 Q2 1

(Sym(Q1 2))

(Sym(Q1 2) (sym(Q2 = 1))

• SES: inserting symmetry breaking constraints (expressed by logic formulae) at the right branch of the binary search tree.

• SBDS is similar to SES. • Good for CSPs with few symmetries• Impractical for CSPs with millions of symmetries. • Short SBDS choose partial symmetries.



SBDD

• Good for solving CSPs with millions of symmetries

• For every CSP, SBDD needs a special dominance checker

1 2

1

1

1 2 3 4

Q1

Q2

Q3

Q4

……

Is current state dominated by previously visited states?



CGT+SBDS/SBDD• CGT: Computational Group Theory• By CGT, a small set of symmetries can g

enerate a huge number of symmetries• CGT+SBDS do not need to explicitly list

all the symmetries (GAP-ECLiPSe SBDS)• CGT+SBDD do not need a specific domi

nance checker any more (GAP-ECLiPSe

SBDD)



Symmetry-breaking heuristic

• dddd



Outline• Definition of a CSP• Motivation of the study of symmetry in CSP• Definitions of symmetry in a CSP• Techniques of symmetry exploitation• Techniques of symmetry detection

– Discrimination tree (neighborhood interchangeability)

– CGRASS– Graph automorphism

• Summary & future research directions



Symmetry detection1. Discrimination tree (to detect NI)

– Before search [Benson and Freuder’92, Haselbock’93]

– During search [Beckwith et al.’01,

Lal and Choueiry’03]

2. CGRASS: for every pair of variables, check whether exchanging these two variables in all occurrences of these two variables does not change the problem [Frisch et al.’02]



Symmetry detection3. First construct a graph

– SAT problem [Crawford’96]– 0-1 ILP problem [Aloul’04]– Problem whose constraints declared in C-like specificati

on language [Ramani and Markov’04]– CSP [Puget’05]

Then use automorphism algorithms: – NAUTY [McKay’81] – SAUCY [Darga et al.’04]– AUTOM [Puget’05]







Summary• Definitions of symmetry• Techniques of exploiting symmetries

– Bundling technique (group NI values together)

– Symmetry breaking (Only visit one of the symmetrical states)

• Static or dynamic• Techniques of detecting symmetries

– Discrimination tree– Graph construction + graph automorphism



Future Research Directions• Lattice domain CSPs

[Gervet’97]

• Exploit symmetries in sub CSPs after structural decomposition of a CSP

• Propagation in non-binary CSPs• Dynamically detect symmetries (beside

s neighborhood interchangeability) during search



Future Research Directions• Develop criteria for systematically

selecting subsets of static symmetry breaking constraints

• Study whether degree-maximization xxx can also be integrated with symmetry-breaking heuristic

• Investigate the possibility of exploiting symmetries for a heuristic function (such as in local search)



Thank you • Thank you for your time and patience to

– read my paper and – attend my presentation

• Questions?



Additional Slides



Computation Group Theory (CGT)Geometric symmetry of 2 2 square• Two symmetries p1= [3, 1, 4, 2] (denote a rotat

ion by 90) and p2= [3, 4, 1, 2] (denote a flip through horizontal axis) can be used to generate other 5 symmetries. For example,

• p1◦p1◦p1 = [4, 3, 2, 1] (rotation by 270)• p2 ◦p1 = [4, 2, 3, 1] (rotation by 90 and then filp)• p1◦p1◦p2 = [2, 1, 4, 3] (flip through vertical axis) ……



Bundling Techniques• Use discrimination tree to find neighborhood i

nterchangeable values [Freuder’91]bluered

white

bluegreenyellow

bluegreenyellow

blue

W Y

ZX

blue|X

green|X

yellow|X

green|Y

yellow|Y

green|X

yellow|X

green|Y

yellow|Y

{red, white}

{blue}

• Joint discrimination tree [Choueiry and Noubir’98]• Non-binary discrimination tree [Lal and Choueiry’03]



Detecting symmetries • Example of graph construction for a SAT

problem [Crawford’96]

C1

x1 x2 x3

I1 I2

C2 C3

x1’ x3’

(x1, x2, x3)= (x1 x2 x3) (x1 x4) (x3 x4)Implied clauses: x1 x1 x3 x3

x4



Detecting symmetries• Example of graph construction for a 0-1 I

LP problem [Aloul et al.’04]

x1 x2 x3 (x1, x2, x3)= P1: 2 x1 + x2 + x3 2 P2: x1 + 2 x2 + x3 2

Y2 Y1

X2,1 X1,1

X1,2 X2,2



Detecting symmetries• Example of graph construction for a C-like spec

ification language [Ramani and Markov’04]

int3 x1, x2;x1*x1 + x2*x2 == 25;x1 >= 1;x2 >= 1;

25

==

+

*

x1

>=

1

*

x2

>=

1



Detecting symmetries• Example of graph construction for an extensiona

l constraint Cxy = {(1, 2), (1, 3), (2, 3)} [Puget’05]

2

1

A3

A2

Cxy

3

x

A1

2

1

3

y



Detecting symmetries• Example of graph construction for two in

tensional constraints [Puget’05]

< d

x

10y

+

x y z

=

x < y x + y + z = 10

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Survey of Techniques for Detecting and Exploiting Symmetry in Constraint Satisfaction Problems