Symmetry Definitions for Constraint Satisfaction Problems

  • Authors:

  • David Cohen;Peter Jeavons;Christopher Jefferson;Karen E. Petrie;Barbara M. Smith

  • Affiliations:

  • Department of Computer Science, Royal Holloway, University of London, London, UK;Computing Laboratory, University of Oxford, Oxford, UK;School of Computer Science, University of St Andrews, Scotland, UK;School of Computer Science, University of St Andrews, Scotland, UK;Cork Constraint Computation Centre, University College Cork, Cork, Ireland

  • Venue:
  • Constraints
  • Year:
  • 2006

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Abstract

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.