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Artificial Intelligence
Volume 156, Issue 2, July 2004, Pages 177-196
 
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doi:10.1016/j.artint.2004.02.003    How to Cite or Link Using DOI (Opens New Window)
Copyright © 2004 Elsevier B.V. All rights reserved.

The complexity of constraint satisfaction problems for small relation algebras

The problems that deserve an attack demonstrate it by a counterattack. Paul Erdös

M. CristaniE-mail The Corresponding Author, a and R. HirschCorresponding Author Contact Information, E-mail The Corresponding Author, b

a Dipartimento di Informatica, Università di Verona, Cà Vignal 2, strada Le Grazie 15, I-37134, Verona, Italy b Department of Computer Science, University College London, Gower Street, London, WC1E 6BT, UK

Received 6 June 2003. 
Available online 25 March 2004.

Abstract

Andréka and Maddux [Notre Dame J. Formal Logic 35 (4) 1994] classified the small relation algebras—those with at most 8 elements, or in other terms, at most 3 atomic relations. They showed that there are eighteen isomorphism types of small relation algebras, all representable. For each simple, small relation algebra they computed the spectrum of the algebra, namely the set of cardinalities of square representations of that relation algebra.

In this paper we analyze the computational complexity of the problem of deciding the satisfiability of a finite set of constraints built on any small relation algebra. We give a complete classification of the complexities of the general constraint satisfaction problem for small relation algebras. For three of the small relation algebras the constraint satisfaction problem is NP-complete, for the other fifteen small relation algebras the constraint satisfaction problem has cubic (or lower) complexity.

We also classify the complexity of the constraint satisfaction problem over fixed finite representations of any relation algebra. If the representation has size two or less then the complexity is cubic (or lower), but if the representation is square, finite and bigger than two then the complexity is NP-complete.

Author Keywords: Relation algebra; Constraint satisfaction problem; Computational complexity

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