Towards a dichotomy theorem for the counting constraint satisfaction problem

The Counting Constraint Satisfaction Problem (#CSP) can be expressed as follows: given a set of variables, a set of values that can be taken by the variables, and a set of constraints specifying some restrictions on the values that can be taken simultaneously by some variables, determine the number...

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Detalhes bibliográficos
Autores: Bulatov, Andrei A., Dalmau, Víctor
Formato: artículo
Estado:Versión aceptada para publicación
Fecha de publicación:2007
País:España
Recursos:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repositorio:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:10230/36327
Acesso em linha:http://hdl.handle.net/10230/36327
http://dx.doi.org/10.1016/j.ic.2006.09.005
Access Level:acceso abierto
Palavra-chave:Constraint satisfaction problem
Counting problems
Complexity
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spelling Towards a dichotomy theorem for the counting constraint satisfaction problemBulatov, Andrei A.Dalmau, VíctorConstraint satisfaction problemCounting problemsComplexityThe Counting Constraint Satisfaction Problem (#CSP) can be expressed as follows: given a set of variables, a set of values that can be taken by the variables, and a set of constraints specifying some restrictions on the values that can be taken simultaneously by some variables, determine the number of assignments of values to variables that satisfy all the constraints. The #CSP provides a general framework for numerous counting combinatorial problems including counting satisfying assignments to a propositional formula, counting graph homomorphisms, graph reliability and many others. This problem can be parametrized by the set of relations that may appear in a constraint. In this paper we start a systematic study of subclasses of the #CSP restricted in this way. The ultimate goal of this investigation is to distinguish those restricted subclasses of the #CSP which are solvable in polynomial time from those which are not. We show that the complexity of any restricted #CSP class on a finite domain can be deduced from the properties of polymorphisms of the allowed constraints, similar to that for the decision constraint satisfaction problem. Then we prove that if a subclass of the #CSP is solvable in polynomial time, then constraints allowed by the class satisfy some very restrictive condition: they need to have a Mal’tsev polymorphism, that is a ternary operation such that . This condition uniformly explains many existing complexity results for particular cases of the #CSP, including the dichotomy results for the problem of counting graph homomorphisms, and it allows us to obtain new results.Elsevier201920192007info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionapplication/pdfapplication/pdfhttp://hdl.handle.net/10230/36327http://dx.doi.org/10.1016/j.ic.2006.09.005reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésInformation and Computation. 2007 May;205(5):651-78.© Elsevier http://dx.doi.org/10.1016/j.ic.2006.09.005info:eu-repo/semantics/openAccessoai:recercat.cat:10230/363272026-05-29T05:05:01Z
dc.title.none.fl_str_mv Towards a dichotomy theorem for the counting constraint satisfaction problem
title Towards a dichotomy theorem for the counting constraint satisfaction problem
spellingShingle Towards a dichotomy theorem for the counting constraint satisfaction problem
Bulatov, Andrei A.
Constraint satisfaction problem
Counting problems
Complexity
title_short Towards a dichotomy theorem for the counting constraint satisfaction problem
title_full Towards a dichotomy theorem for the counting constraint satisfaction problem
title_fullStr Towards a dichotomy theorem for the counting constraint satisfaction problem
title_full_unstemmed Towards a dichotomy theorem for the counting constraint satisfaction problem
title_sort Towards a dichotomy theorem for the counting constraint satisfaction problem
dc.creator.none.fl_str_mv Bulatov, Andrei A.
Dalmau, Víctor
author Bulatov, Andrei A.
author_facet Bulatov, Andrei A.
Dalmau, Víctor
author_role author
author2 Dalmau, Víctor
author2_role author
dc.subject.none.fl_str_mv Constraint satisfaction problem
Counting problems
Complexity
topic Constraint satisfaction problem
Counting problems
Complexity
description The Counting Constraint Satisfaction Problem (#CSP) can be expressed as follows: given a set of variables, a set of values that can be taken by the variables, and a set of constraints specifying some restrictions on the values that can be taken simultaneously by some variables, determine the number of assignments of values to variables that satisfy all the constraints. The #CSP provides a general framework for numerous counting combinatorial problems including counting satisfying assignments to a propositional formula, counting graph homomorphisms, graph reliability and many others. This problem can be parametrized by the set of relations that may appear in a constraint. In this paper we start a systematic study of subclasses of the #CSP restricted in this way. The ultimate goal of this investigation is to distinguish those restricted subclasses of the #CSP which are solvable in polynomial time from those which are not. We show that the complexity of any restricted #CSP class on a finite domain can be deduced from the properties of polymorphisms of the allowed constraints, similar to that for the decision constraint satisfaction problem. Then we prove that if a subclass of the #CSP is solvable in polynomial time, then constraints allowed by the class satisfy some very restrictive condition: they need to have a Mal’tsev polymorphism, that is a ternary operation such that . This condition uniformly explains many existing complexity results for particular cases of the #CSP, including the dichotomy results for the problem of counting graph homomorphisms, and it allows us to obtain new results.
publishDate 2007
dc.date.none.fl_str_mv 2007
2019
2019
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/10230/36327
http://dx.doi.org/10.1016/j.ic.2006.09.005
url http://hdl.handle.net/10230/36327
http://dx.doi.org/10.1016/j.ic.2006.09.005
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Information and Computation. 2007 May;205(5):651-78.
dc.rights.none.fl_str_mv © Elsevier http://dx.doi.org/10.1016/j.ic.2006.09.005
info:eu-repo/semantics/openAccess
rights_invalid_str_mv © Elsevier http://dx.doi.org/10.1016/j.ic.2006.09.005
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
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