1 Constraint Symmetry and Solution Symmetry Presented by Beau M. Christ Symmetry in CSP’s Spring...

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Constraint Constraint Symmetry andSymmetry and

Solution SymmetrySolution Symmetry

Constraint Constraint Symmetry andSymmetry and

Solution SymmetrySolution Symmetry

Presented by Beau M. ChristPresented by Beau M. Christ

Symmetry in CSP’s Spring 2010Symmetry in CSP’s Spring 2010Presented by Beau M. ChristPresented by Beau M. Christ

Symmetry in CSP’s Spring 2010Symmetry in CSP’s Spring 2010

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OutlineOutlineOutlineOutlineIntroductionIntroduction

DefinitionsDefinitions

Constraint and Solution Symmetry Constraint and Solution Symmetry RelationshipsRelationships

Symmetry in CSPs with Few SolutionsSymmetry in CSPs with Few Solutions

Symmetry in SATSymmetry in SAT

Symmetry in PracticeSymmetry in Practice

ConclusionConclusion

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IntroductionIntroductionIntroductionIntroduction

Symmetry can be problematicSymmetry can be problematic

Thus, symmetry-breaking research is Thus, symmetry-breaking research is

activeactive

And yet, how do we define symmetry?And yet, how do we define symmetry?

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Definition #1Definition #1Definition #1Definition #1

A A CSP instanceCSP instance is a triple ⟨V,D,C⟩ where: is a triple ⟨V,D,C⟩ where:• V is a set of variables;V is a set of variables;• D is a universal domain, specifying the D is a universal domain, specifying the possible values for those variables;possible values for those variables;• C is a set of constraints. Each constraint c ∈ C C is a set of constraints. Each constraint c ∈ C is a pair c = ⟨σ,ρ⟩ where σ is a list of variables is a pair c = ⟨σ,ρ⟩ where σ is a list of variables from V, called the from V, called the constraint scopeconstraint scope, and ρ is a , and ρ is a ∣σ∣-ary relation over D, called the ∣σ∣-ary relation over D, called the constraint constraint relation.relation.

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DefinitionsDefinitionsDefinitionsDefinitions

An An assignment of values to variablesassignment of values to variables is a is a set {⟨vset {⟨v11,a,a11⟩,⟨v⟩,⟨v22,a,a22⟩, ..., ⟨v⟩, ..., ⟨vkk,a,akk⟩} where {v⟩} where {v11,v,v22, ..., , ..., vvkk} ⊆ V and a} ⊆ V and aii ∈ D, for all 1 ≤ i ≤ k. ∈ D, for all 1 ≤ i ≤ k.

A A solution to a CSP instancesolution to a CSP instance ⟨V,D,C⟩ ⟨V,D,C⟩ is a is a mapping from V into D whose restriction to mapping from V into D whose restriction to each constraint scope is in the corresponding each constraint scope is in the corresponding constraint relation, i.e., is allowed by the constraint relation, i.e., is allowed by the constraintconstraint

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DefinitionsDefinitionsDefinitionsDefinitionsA general agreement for A general agreement for symmetry in CSPssymmetry in CSPs

Acts on variable-value pairsActs on variable-value pairs

Symmetries map solutions to solutions Symmetries map solutions to solutions

and non-solutions to non-solutionsand non-solutions to non-solutions

But should any bijective mapping that But should any bijective mapping that

preserves solutions be considered a preserves solutions be considered a

symmetry, or just a consequence of leaving symmetry, or just a consequence of leaving

the constraints unchanged?the constraints unchanged?

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For any CSP instance P = ⟨V,D,C⟩, a For any CSP instance P = ⟨V,D,C⟩, a solution solution symmetrysymmetry of P is a permutation of the set V × D of P is a permutation of the set V × D that preserves the set of solutions to P.that preserves the set of solutions to P.

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But we do not want to have to examine all But we do not want to have to examine all solutions to a problem in order to identify solutions to a problem in order to identify symmetry.symmetry.

This leads us to the second viewpoint of This leads us to the second viewpoint of symmetry, which uses the microstructure of the symmetry, which uses the microstructure of the CSP.CSP.

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For any CSP instance P = ⟨V,D,C⟩, the For any CSP instance P = ⟨V,D,C⟩, the microstructure complement of Pmicrostructure complement of P is a is a hypergraph with set of vertices V× D. A set of hypergraph with set of vertices V× D. A set of vertices is a hyperedge of the microstructure vertices is a hyperedge of the microstructure complement if it represents an assignment complement if it represents an assignment disallowed by a constraint, or consists of a pair disallowed by a constraint, or consists of a pair of incompatible assignments for the same of incompatible assignments for the same variable.variable.

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Note: The vertices of the microstructure Note: The vertices of the microstructure complement are variable-value pairs of the complement are variable-value pairs of the CSP, and a set of vertices CSP, and a set of vertices {⟨v{⟨v11,a,a11⟩,⟨v⟩,⟨v22,a,a22⟩, ..., ⟩, ..., ⟨v⟨vkk,a,akk⟩} is a hyper edge if and only if:⟩} is a hyper edge if and only if:

• {v{v11,v,v22, ..., v, ..., vkk} is the set of variables in the } is the set of variables in the scope of some constraint, but the constraint scope of some constraint, but the constraint disallows the assignment {⟨vdisallows the assignment {⟨v11,a,a11⟩,⟨v⟩,⟨v22,a,a22⟩, ..., ⟩, ..., ⟨v⟨vkk,a,akk⟩}; or⟩}; or• k=2, vk=2, v11 = v = v22, and a, and a11 ≠ a ≠ a22

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For any CSP instance P = For any CSP instance P = ⟨V,D,C⟩, a ⟨V,D,C⟩, a constraint symmetryconstraint symmetry is an automorphism of is an automorphism of the microstructure complement of P (or, the microstructure complement of P (or, equivalently, of the microstructure).equivalently, of the microstructure).

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Example #1: n-Example #1: n-queensqueens


Standard formulationStandard formulation

V = {rV = {r11,r,r22, ..., r, ..., rnn} representing the rows} representing the rows

D = {1,2, ..., n} representing the columnsD = {1,2, ..., n} representing the columns

C (intension)C (intension)

for all i, j, 1 ≤ i < j ≤ n, rfor all i, j, 1 ≤ i < j ≤ n, rii ≠ r ≠ rj j ;;

for all i, j, 1 ≤ i < j ≤ n, ∣rfor all i, j, 1 ≤ i < j ≤ n, ∣rii - r - rjj∣ ≠ ∣i - j∣∣ ≠ ∣i - j∣

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Chessboard has 8 symmetries (x, y, d1, Chessboard has 8 symmetries (x, y, d1,

d2, r90, r180, r270, id)d2, r90, r180, r270, id)

Symmetries map vvp ⟨rSymmetries map vvp ⟨rii,k⟩ to ,k⟩ to

(respectively):(respectively):

⟨r⟨rn+1-in+1-i,k⟩ , ⟨r,k⟩ , ⟨rii,n+1-k⟩ , ⟨r,n+1-k⟩ , ⟨rii,i⟩ , ⟨r,i⟩ , ⟨rn+1-kn+1-k,n+1-i⟩ ,,n+1-i⟩ ,

⟨r⟨rkk,n+1-i⟩ , ⟨r,n+1-i⟩ , ⟨rn+1-in+1-i,n+1-k⟩ , ⟨r,n+1-k⟩ , ⟨rn+1-kn+1-k,i⟩ , ⟨r,i⟩ , ⟨rii,k⟩,k⟩

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Example 1: n-Example 1: n-queensqueens

Example 1: n-Example 1: n-queensqueens

This hides symmetry between rows and This hides symmetry between rows and

columnscolumns

This symmetry is restored using the This symmetry is restored using the

microstructure complement, as it treats microstructure complement, as it treats

both reasons for a disallowed pair of both reasons for a disallowed pair of

assignments equallyassignments equally

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Relationships: Relationships: Theorem 1Theorem 1


The group of constraint symmetries of a The group of constraint symmetries of a CSP instance P CSP instance P is a subgroup ofis a subgroup of the group the group of solution symmetries of P.of solution symmetries of P.

Note: The converse is not true.Note: The converse is not true.

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Relationships: Definition Relationships: Definition #5#5

Relationships: Definition Relationships: Definition #5#5

For any CSP instance P, a k-ary nogood is For any CSP instance P, a k-ary nogood is an assignment to k variables of P that an assignment to k variables of P that cannot be extended to a solution of P.cannot be extended to a solution of P.

The k-nogood hypergraph of P is a The k-nogood hypergraph of P is a hypergraph whose set of vertices is V× D hypergraph whose set of vertices is V× D and whose set of edges is the set of all m-and whose set of edges is the set of all m-ary nogoods for all m ≤ k.ary nogoods for all m ≤ k.

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For any k-ary CSP instance P, the group of For any k-ary CSP instance P, the group of all solution symmetries of P all solution symmetries of P is equal tois equal to the automorphism group of the k-nogood the automorphism group of the k-nogood hypergraph of P.hypergraph of P.

Thus, the k-nogood hypergraph of a CSP Thus, the k-nogood hypergraph of a CSP instance has the same vertices as the instance has the same vertices as the microstructure component.microstructure component.

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Example 2: 4-Example 2: 4-queensqueens


Fig. 1: The 4-queens solutions and the complementFig. 1: The 4-queens solutions and the complementof the binary no-good hypergraphof the binary no-good hypergraph

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The automorphisms are:The automorphisms are:

vertices within either clique can be vertices within either clique can be

permutedpermuted

vertices in one clique can be swapped vertices in one clique can be swapped

with the otherwith the other

eight isolated vertices can be permutedeight isolated vertices can be permuted

compose these permutationscompose these permutations

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Since the automorphisms of a Since the automorphisms of a graph are the same as the graph are the same as the automorphisms of its automorphisms of its complement, then theorem 2 complement, then theorem 2 tells us these are the solution tells us these are the solution symmetries for the 4-queens symmetries for the 4-queens problem.problem.

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CSPs with Few CSPs with Few SolutionsSolutions


Any permutation of vvp’s is a solution Any permutation of vvp’s is a solution symmetry when a CSP instance has no symmetry when a CSP instance has no solutions.solutions.

Furthermore, CSP instances with very few Furthermore, CSP instances with very few solutions must have many solution solutions must have many solution symmetries.symmetries.

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Example 3Example 3Example 3Example 3

Suppose a CSP instance with n variables and d Suppose a CSP instance with n variables and d

values for each variable has only one solutionvalues for each variable has only one solution

Any permutation of the n vvp’s in the solution Any permutation of the n vvp’s in the solution

is a solution symmetryis a solution symmetry

The permutations of the n(d-1) vvp’s not in The permutations of the n(d-1) vvp’s not in

the solution are also solution symmetriesthe solution are also solution symmetries

Thus, the solution symmetry group has n! × Thus, the solution symmetry group has n! ×

(n(d-1))! elements(n(d-1))! elements

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Suppose we have a CSP instance with 2 Suppose we have a CSP instance with 2

solutions, where each solution has k solutions, where each solution has k

assignments in commonassignments in common

A solution symmetry must permute the k A solution symmetry must permute the k

common assignments amongst themselvescommon assignments amongst themselves

The remaining (n-k) assignments in each The remaining (n-k) assignments in each

solution can also be permutedsolution can also be permuted

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The two solutions can be swappedThe two solutions can be swapped

The other (nd-2n+k) assignments can also The other (nd-2n+k) assignments can also

be permuted.be permuted.

Thus, we have k! × (n-k)!Thus, we have k! × (n-k)!2 2 × 2 × (nd-× 2 × (nd-

2n+k)! solution symmetries2n+k)! solution symmetries

Note: This is 4-queens with k=0Note: This is 4-queens with k=0

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The basic idea is that it is often more The basic idea is that it is often more

difficult to construct permutations with difficult to construct permutations with

more solutionsmore solutions

For n-queens where n >6, it appears that For n-queens where n >6, it appears that

the solution symmetries are exactly the 8 the solution symmetries are exactly the 8

symmetries of the constraint symmetry symmetries of the constraint symmetry

groupgroup

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One way of dealing with constraint One way of dealing with constraint

symmetry is to add symmetry-breaking symmetry is to add symmetry-breaking

constraintsconstraints

Symmetry breaking can eliminate Symmetry breaking can eliminate

constraint symmetry but lead to a much constraint symmetry but lead to a much

larger solution symmetry grouplarger solution symmetry group

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Symmetry in SATSymmetry in SATSymmetry in SATSymmetry in SAT

A SAT instance can be viewed as a CSP A SAT instance can be viewed as a CSP

instanceinstance

We can define a symmetry of a SAT We can define a symmetry of a SAT

instance as a permutation of the variables instance as a permutation of the variables

that leaves the set of clauses unchangedthat leaves the set of clauses unchanged

(Aloul et al. 2003) allow symmetries to act (Aloul et al. 2003) allow symmetries to act

on literals, and thus is similar to our on literals, and thus is similar to our

definition of constraint symmetrydefinition of constraint symmetry

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Symmetry in SATSymmetry in SATSymmetry in SATSymmetry in SAT

They construct a graph to represent the They construct a graph to represent the

set of clauses, whose automorphisms are set of clauses, whose automorphisms are

the symmetries of the SAT instancethe symmetries of the SAT instance

The graph is similar to the microstructure The graph is similar to the microstructure

complement, but has vertices for clauses complement, but has vertices for clauses

as well as literals, and colors for different as well as literals, and colors for different

types of vertextypes of vertex

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Symmetry in Symmetry in PracticePractice

Symmetry in Symmetry in PracticePractice

Identifying symmetry helps ease searchIdentifying symmetry helps ease search

Add symmetry breaking constraints OR use Add symmetry breaking constraints OR use

a dynamic method that adds symmetry a dynamic method that adds symmetry

breaking constraints upon backtrackingbreaking constraints upon backtracking

In practice, researchers identify constraint In practice, researchers identify constraint

symmetries rather than solution symmetries rather than solution

symmetriessymmetries

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InterchangeabilityInterchangeabilityInterchangeabilityInterchangeability

Freuder’s interchangeability is an explicit Freuder’s interchangeability is an explicit

form of solution symmetryform of solution symmetry

Full interchangeability can determine Full interchangeability can determine

values that are, in a sense, equivalentvalues that are, in a sense, equivalent

This, in general, requires finding all This, in general, requires finding all

solutionssolutions

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InterchangeabilityInterchangeabilityInterchangeabilityInterchangeability

Freuder defines local forms of Freuder defines local forms of

interchangeability (such as neighborhood interchangeability (such as neighborhood

interchangeability, which is a form of interchangeability, which is a form of

constraint symmetry)constraint symmetry)

Future work aims to identify values that Future work aims to identify values that

are interchangeable in the set of solutions, are interchangeable in the set of solutions,

and not just the constraintsand not just the constraints

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ConclusionConclusionConclusionConclusion

Constraint symmetry: automorphism groups of the Constraint symmetry: automorphism groups of the

microstructure complementmicrostructure complement

Solution symmetry: automorphism groups of the k-ary Solution symmetry: automorphism groups of the k-ary

nogood hypergraphnogood hypergraph

Solution symmetry may be less useful than constraint Solution symmetry may be less useful than constraint

symmetrysymmetry

Although most research has dealt with constraint Although most research has dealt with constraint

symmetry, new research is being done with symmetry, new research is being done with

interchangeabilityinterchangeability

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ReferenceReferenceReferenceReference

““Constraint Symmetry and Solution Constraint Symmetry and Solution Symmetry”Symmetry”

---Cohen et. al------Cohen et. al---AAAI 2006AAAI 2006

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That’s all, folks!That’s all, folks!That’s all, folks!That’s all, folks!

Questions?Questions?

Comments?Comments?

Remarks?Remarks?

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