Independent sets from an algebraic perspective

Autores
Dickenstein, Alicia Marcela; Tobis, Enrique Augusto
Año de publicación
2012
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In this paper, we study the basic problem of counting independent sets in a graph and, in particular, the problem of counting antichains in a finite poset, from an algebraic perspective. We show that neither independence polynomials of bipartite Cohen–Macaulay graphs nor Hilbert series of initial ideals of radical zero-dimensional complete intersections ideals, can be evaluated in polynomial time, unless #P = P. Moreover, we present a family of radical zero-dimensional complete intersection ideals JP associated to a finite poset P, for which we describe a universal Gröbner basis. This implies that the bottleneck in computing the dimension of the quotient by JP (that is, the number of zeros of JP) using Gröbner methods lies in the description of the standard monomials.
Fil: Dickenstein, Alicia Marcela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Tobis, Enrique Augusto. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Materia
Ideal
Hilbert Function
Polynomial Time
Groebner Basis
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/19930
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/19930
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Independent sets from an algebraic perspectiveDickenstein, Alicia MarcelaTobis, Enrique AugustoIdealHilbert FunctionPolynomial TimeGroebner Basishttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1In this paper, we study the basic problem of counting independent sets in a graph and, in particular, the problem of counting antichains in a finite poset, from an algebraic perspective. We show that neither independence polynomials of bipartite Cohen–Macaulay graphs nor Hilbert series of initial ideals of radical zero-dimensional complete intersections ideals, can be evaluated in polynomial time, unless #P = P. Moreover, we present a family of radical zero-dimensional complete intersection ideals JP associated to a finite poset P, for which we describe a universal Gröbner basis. This implies that the bottleneck in computing the dimension of the quotient by JP (that is, the number of zeros of JP) using Gröbner methods lies in the description of the standard monomials.Fil: Dickenstein, Alicia Marcela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Tobis, Enrique Augusto. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaWorld Scientific2012-03info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/19930Dickenstein, Alicia Marcela; Tobis, Enrique Augusto; Independent sets from an algebraic perspective; World Scientific; International Journal of Algebra and Computation; 22; 2; 3-2012; 14-290218-19671793-6500CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1142/S0218196711006819info:eu-repo/semantics/altIdentifier/url/http://www.worldscientific.com/doi/abs/10.1142/S0218196711006819info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1003.3508info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T10:01:32Zoai:ri.conicet.gov.ar:11336/19930instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-03 10:01:32.638CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Independent sets from an algebraic perspective
title Independent sets from an algebraic perspective
spellingShingle Independent sets from an algebraic perspective
Dickenstein, Alicia Marcela
Ideal
Hilbert Function
Polynomial Time
Groebner Basis
title_short Independent sets from an algebraic perspective
title_full Independent sets from an algebraic perspective
title_fullStr Independent sets from an algebraic perspective
title_full_unstemmed Independent sets from an algebraic perspective
title_sort Independent sets from an algebraic perspective
dc.creator.none.fl_str_mv Dickenstein, Alicia Marcela
Tobis, Enrique Augusto
author Dickenstein, Alicia Marcela
author_facet Dickenstein, Alicia Marcela
Tobis, Enrique Augusto
author_role author
author2 Tobis, Enrique Augusto
author2_role author
dc.subject.none.fl_str_mv Ideal
Hilbert Function
Polynomial Time
Groebner Basis
topic Ideal
Hilbert Function
Polynomial Time
Groebner Basis
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv In this paper, we study the basic problem of counting independent sets in a graph and, in particular, the problem of counting antichains in a finite poset, from an algebraic perspective. We show that neither independence polynomials of bipartite Cohen–Macaulay graphs nor Hilbert series of initial ideals of radical zero-dimensional complete intersections ideals, can be evaluated in polynomial time, unless #P = P. Moreover, we present a family of radical zero-dimensional complete intersection ideals JP associated to a finite poset P, for which we describe a universal Gröbner basis. This implies that the bottleneck in computing the dimension of the quotient by JP (that is, the number of zeros of JP) using Gröbner methods lies in the description of the standard monomials.
Fil: Dickenstein, Alicia Marcela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Tobis, Enrique Augusto. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
description In this paper, we study the basic problem of counting independent sets in a graph and, in particular, the problem of counting antichains in a finite poset, from an algebraic perspective. We show that neither independence polynomials of bipartite Cohen–Macaulay graphs nor Hilbert series of initial ideals of radical zero-dimensional complete intersections ideals, can be evaluated in polynomial time, unless #P = P. Moreover, we present a family of radical zero-dimensional complete intersection ideals JP associated to a finite poset P, for which we describe a universal Gröbner basis. This implies that the bottleneck in computing the dimension of the quotient by JP (that is, the number of zeros of JP) using Gröbner methods lies in the description of the standard monomials.
publishDate 2012
dc.date.none.fl_str_mv 2012-03
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/19930
Dickenstein, Alicia Marcela; Tobis, Enrique Augusto; Independent sets from an algebraic perspective; World Scientific; International Journal of Algebra and Computation; 22; 2; 3-2012; 14-29
0218-1967
1793-6500
CONICET Digital
CONICET
url http://hdl.handle.net/11336/19930
identifier_str_mv Dickenstein, Alicia Marcela; Tobis, Enrique Augusto; Independent sets from an algebraic perspective; World Scientific; International Journal of Algebra and Computation; 22; 2; 3-2012; 14-29
0218-1967
1793-6500
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1142/S0218196711006819
info:eu-repo/semantics/altIdentifier/url/http://www.worldscientific.com/doi/abs/10.1142/S0218196711006819
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1003.3508
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv World Scientific
publisher.none.fl_str_mv World Scientific
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
_version_ 1842269702574833664
score 13.13397

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