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Journal of Logic and Computation 2006 16(3):359-373; doi:10.1093/logcom/exi083
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Vol. 16 No. 3, © The Author, 2006. Published by Oxford University Press. All rights reserved.

Original Articles

Constraint Satisfaction with Countable Homogeneous Templates

Manuel Bodirsky1 and Jaroslav Nesetril2

1 Institut für Informatik, Humboldt-Universität zu Berlin, Germany. Email: bodirsky{at}informatik.hu-berlin.de, 2 Institute for Theoretical Computer Science (ITI), and Department of Applied Mathematics (KAM), Charles University, Prague, Czech Republic. Email: nesetril{at}kam.mff.cuni.cz

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

Keywords: Complexity of constraint satisfaction, homogeneous digraphs, graph homomorphisms, {omega}-categorical structures, polymorphism preservation theorem

Received 4 April 2005.


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