Constraint representations and structural tractability

  • Authors:

  • David A. Cohen;Martin J. Green;Chris Houghton

  • Affiliations:

  • Department of Computer Science, Royal Holloway, University of London, UK;Department of Computer Science, Royal Holloway, University of London, UK;Department of Computer Science, Royal Holloway, University of London, UK

  • Venue:
  • CP'09 Proceedings of the 15th international conference on Principles and practice of constraint programming
  • Year:
  • 2009

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Abstract

The intractability of the general CSP has motivated the search for restrictions which lead to tractable fragments. One way to achieve tractability is to restrict the structure of the instances. As much of the work in this area arises from similar work on databases it has been a natural assumption that all constraint relations are explicitly represented. If this is the case then all instances with an acyclic hypergraph structure are tractable. Unfortunately this result does not hold if we are allowed to represent constraint relations implicitly: the class of SAT instances with acyclic hypergraph structure is NP-hard. Continuing the work of Chen and Grohe on the succinct GDNF representation we develop the theory of structural tractability for an extension to the table constraint that has a succinct representation of SAT clauses. This mixed representation is less succinct than the GDNF representation but more succinct than the table representation. We prove a strict hierarchy of structural tractability for the GDNF, the mixed, and the explicit representations of constraint relations. Using this proof we are able to show that the mixed representation provides novel tractable structural classes. Since the mixed representation naturally extends SAT, this provides a useful result, extending known structural tractability results for SAT. Under a natural restriction we are able precisely to capture the tractable structural classes for this mixed representation. This gives us an extension of Grohe's dichotomy theorem for the tractability of classes of relational structures with a fixed signature. In particular it captures the tractability of some classes of unbounded arity, specifically the class of CSPs with precisely one constraint.