Constraint Satisfaction with Countable Homogeneous Templates

  • Authors:

  • Manuel Bodirsky;Jaroslav Nešetřil

  • Affiliations:

  • Institut für Informatik, Humboldt-Universität zu Berlin, Germany. Email: bodirsky@informatik.hu-berlin.de,;Institute for Theoretical Computer Science (ITI), and Department of Applied Mathematics (KAM), Charles University, Prague, Czech Republic. Email: nesetril@kam.mff.cuni.cz

  • Venue:
  • Journal of Logic and Computation
  • Year:
  • 2006

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Abstract

For a fixed countable homogeneous relational structure Γ we study the computational problem whether a given finite structure of the same signature homomorphically maps to Γ. This problem is known as the constraint satisfaction problem CSP(Γ) for the template Γ and has been intensively studied for finite Γ. We show that --- as in the case of finite Γ --- the computational complexity of CSP(Γ) for countable homogeneous Γ is determined by the clone of polymorphisms of Γ. To this end we prove the following theorem, which is of independent interest: the primitive positive definable relations over an ω-categorical structure Γ are precisely the relations that are preserved by the polymorphisms of Γ. If the age of Γ is given by a finite number of finite forbidden induced substructures, then CSP(Γ) is in NP. We use a classification result by Cherlin and prove that in this case every constraint satisfaction problem for a countable homogeneous digraph is either tractable or NP-complete.