Principles and Practice of Constraint Programming - CP 2005

Volume 3709 of the series Lecture Notes in Computer Science pp 17-31

Symmetry Definitions for Constraint Satisfaction Problems

  • David CohenAffiliated withDepartment of Computer Science, Royal Holloway, University of London
  • , Peter JeavonsAffiliated withComputing Laboratory, University of Oxford
  • , Christopher JeffersonAffiliated withDepartment of Computer Science, University of York
  • , Karen E. PetrieAffiliated withCork Constraint Computation Centre, University College Cork
  • , Barbara M. SmithAffiliated withCork Constraint Computation Centre, University College Cork

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We review the many different definitions of symmetry for constraint satisfaction problems (CSPs) that have appeared in the literature, and show that a symmetry can be defined in two fundamentally different ways: as an operation preserving the solutions of a CSP instance, or else as an operation preserving the constraints. We refer to these as solution symmetries and constraint symmetries. We define a constraint symmetry more precisely as an automorphism of a hypergraph associated with a CSP instance, the microstructure complement. We show that the solution symmetries of a CSP instance can also be obtained as the automorphisms of a related hypergraph, the k-ary nogood hypergraph and give examples to show that some instances have many more solution symmetries than constraint symmetries. Finally, we discuss the practical implications of these different notions of symmetry.