Constraint Satisfaction with Countable Homogeneous Templates

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...

• Type: Conference
• Year: 2003
• Where: CSL
• Authors: Manuel Bodirsky, Jaroslav Nesetril
• Keywords: Computational Problem, Constraint Satisfaction Problem, CSL 2003, Homogeneous Relational Structure