Symmetry Definitions for Constraint Satisfaction Problems Dave Cohen, Peter Jeavons, Chris...

Symmetry Definitions for Symmetry Definitions for Constraint Satisfaction ProblemsConstraint Satisfaction Problems

Dave Cohen, Peter Jeavons, Chris Jefferson, Karen Petrie and Barbara Smith

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Symmetry Symmetry • Symmetry in CSPs has been an active

research area for several years• e.g. SymCon workshops at the CP conferences

since 2001

• But researchers define symmetry in CSPs in different ways• are they defining different things?

• A symmetry of a CSP is a transformation of the CSP that leaves some property of the CSP unchanged• what does the symmetry act on? • what property does it leave unchanged?

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• xi =j if the queen in row i is in column j • Symmetries can act on the variables:

• xi =j → x6-i =j

• or the values

• xi =j → xi =6-j

• or both

• xi =j → xj =i

• To cover all these, we define symmetries as acting on assignments, i.e. variable–value pairs

Symmetries of 5−queensSymmetries of 5−queens

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What property is preserved?What property is preserved?

• A symmetry of a CSP P is a permutation of the variable-value pairs that preserves the solutions of P

• A symmetry of a CSP P is a permutation of the variable-value pairs that preserves the constraints of P• and hence also preserves the solutions of P

• These are not equivalent

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What property is preserved?What property is preserved?

• A solution symmetry of a CSP P is a permutation of the variable-value pairs that preserves the solutions of P

• A constraint symmetry of a CSP P is a permutation of the variable-value pairs that preserves the constraints of P• and hence also preserves the solutions of P

• These are not equivalent

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Microstructure ComplementMicrostructure Complement• A hypergraph with

• a vertex for every variable-value pair

• an edge for any pair of vertices representing assignments to the same variable

• a hyperedge for any set of vertices representing a tuple forbidden by a constraint

w,1

w,0

x,0y,0

z,1

z,0

y,1 x,1

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Definition of Constraint SymmetryDefinition of Constraint Symmetry

• An automorphism of a (hyper)graph is a bijection on the vertices that preserves the (hyper)edges

• We define a constraint symmetry as an automorphism of the microstructure complement • (w,0 w,1) (y,0 y,1) (z,0 z,1)• (y,0 z,1) (y,1 z,0)• (w,0 w,1) (y,0 z,0) (y,1 z,1)• identity

• i.e. a bijection on the variable-value pairs that preserves the constraints

w,1

w,0

x,0

y,0

z,1

z,0

y,1

x,1

w,1

w,0

x,0y,0

z,1

z,0

y,1 x,1

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Solution Symmetries and Constraint SymmetriesSolution Symmetries and Constraint Symmetries

• The constraint symmetries of a CSP are a subgroup of the solution symmetries

• There can be many more solution symmetries than constraint symmetries• e.g. if a CSP has no solution, any permutation of the

variable-value pairs is a solution symmetry

• 4-queens has two solutions• we can see what permutations

of the variable-value pairs will preserve the solutions

• any permutation of x1 ,2, x2 ,4, x3 ,1, x4 ,3 is a solution symmetry

• .. and many more

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The k-ary nogood hypergraph The k-ary nogood hypergraph • A k-ary nogood is an assignment to k

variables that cannot be extended to a solution

• The k-ary nogood hypergraph has the same vertices as the microstructure and a (hyper) edge for every m-ary nogood (m ≤ k)

• The solution symmetry group of a k-ary CSP is the automorphism group of the k-ary nogood hypergraph • e.g. in a binary CSP, we need only consider

the binary and unary nogoods to find all the solution symmetries

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Solution Symmetries of 4-queensSolution Symmetries of 4-queens

• The complement of the binary nogood graph has two cliques, one for each solution

• Some automorphisms:• permuting the isolated

vertices • permuting either clique

independently

• There are 46m. solution symmetries• but 8 constraint

symmetries

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ImplicationsImplications• Finding constraint symmetries automatically is

difficult • symmetries depend on how the constraints are

expressed• often the microstructure is far too big to construct

• more compact representations can be used sometimes• checking a proposed constraint symmetry is easier

• don’t need to construct the microstructure

• Finding all solution symmetries automatically seems pointless• we need the solutions!

• Use nogoods found during search? • add them (& symmetric equivalents) to the

microstructure complement • find the new symmetry group • use that during the remaining search

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ConclusionsConclusions• Symmetry in CSPs has been defined in different ways:

• preserving the solutions• preserving the constraints (and therefore the solutions)

• There can be far more solution symmetries than constraint symmetries• we have shown the relationship between them

• To identify symmetries, people often think of transformations that preserve the solutions• …but we think you are identifying constraint symmetry!

• Defining symmetry appropriately is crucial if we want to find symmetries automatically

THE END