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Symmetry Definitions for Constraint Satisfaction Problems Dave Cohen, Peter Jeavons, Chris Jefferson, Karen Petrie and Barbara Smith

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CP05 2 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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CP05 3 x i =j if the queen in row i is in column j Symmetries can act on the variables: x i =j x 6-i =j or the values x i =j x i =6-j or both x i =j x j =i To cover all these, we define symmetries as acting on assignments, i.e. variable– value pairs Symmetries of 5queens

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CP05 4 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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CP05 5 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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CP05 6 Microstructure 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,0 y,0 z,1 z,0 y,1 x,1

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CP05 7 Definition 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,0 y,0 z,1 z,0 y,1 x,1

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CP05 8 Solution 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 x 1,2, x 2,4, x 3,1, x 4,3 is a solution symmetry.. and many more

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CP05 9 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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CP05 10 Solution 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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CP05 11 Implications 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 dont 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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CP05 12 Conclusions 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

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