Tropical polar cones, hypergraph transversals, and mean payoff games

Autores
Allamigeon, Xavier; Gaubert, Stéphane; Katz, Ricardo David
Año de publicación
2011
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a tropical analogue of Farkas lemma, which allows one to check whether a linear inequality is implied by a finite family of linear inequalities. Here, the certificate is a strategy of a mean payoff game. We discuss examples, showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded, and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone.
Fil: Allamigeon, Xavier. Institut National de Recherche en Informatique et en Automatique; Francia
Fil: Gaubert, Stéphane. École Polytechnique; Francia
Fil: Katz, Ricardo David. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura. Instituto de Matemática "Beppo Levi"; Argentina
Materia
MAX-PLUS SEMIRING
MAX-PLUS CONVEXITY
TROPICAL CONVEXITY
POLYHEDRA
HYPERGRAPH TRANSVERSALS
MINIMAL HITTING SETS
MINIMAL SOLUTIONS
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:11336/268274
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/268274
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Tropical polar cones, hypergraph transversals, and mean payoff gamesAllamigeon, XavierGaubert, StéphaneKatz, Ricardo DavidMAX-PLUS SEMIRINGMAX-PLUS CONVEXITYTROPICAL CONVEXITYPOLYHEDRAHYPERGRAPH TRANSVERSALSMINIMAL HITTING SETSMINIMAL SOLUTIONShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a tropical analogue of Farkas lemma, which allows one to check whether a linear inequality is implied by a finite family of linear inequalities. Here, the certificate is a strategy of a mean payoff game. We discuss examples, showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded, and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone.Fil: Allamigeon, Xavier. Institut National de Recherche en Informatique et en Automatique; FranciaFil: Gaubert, Stéphane. École Polytechnique; FranciaFil: Katz, Ricardo David. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura. Instituto de Matemática "Beppo Levi"; ArgentinaElsevier Science Inc.2011-10info: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/11336/268274Allamigeon, Xavier; Gaubert, Stéphane; Katz, Ricardo David; Tropical polar cones, hypergraph transversals, and mean payoff games; Elsevier Science Inc.; Linear Algebra and its Applications; 435; 7; 10-2011; 1549-15740024-3795CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0024379511001042info:eu-repo/semantics/altIdentifier/doi/10.1016/j.laa.2011.02.004info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:31:23Zoai:ri.conicet.gov.ar:11336/268274instacron: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:34982025-09-29 10:31:23.262CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Tropical polar cones, hypergraph transversals, and mean payoff games
title Tropical polar cones, hypergraph transversals, and mean payoff games
spellingShingle Tropical polar cones, hypergraph transversals, and mean payoff games
Allamigeon, Xavier
MAX-PLUS SEMIRING
MAX-PLUS CONVEXITY
TROPICAL CONVEXITY
POLYHEDRA
HYPERGRAPH TRANSVERSALS
MINIMAL HITTING SETS
MINIMAL SOLUTIONS
title_short Tropical polar cones, hypergraph transversals, and mean payoff games
title_full Tropical polar cones, hypergraph transversals, and mean payoff games
title_fullStr Tropical polar cones, hypergraph transversals, and mean payoff games
title_full_unstemmed Tropical polar cones, hypergraph transversals, and mean payoff games
title_sort Tropical polar cones, hypergraph transversals, and mean payoff games
dc.creator.none.fl_str_mv Allamigeon, Xavier
Gaubert, Stéphane
Katz, Ricardo David
author Allamigeon, Xavier
author_facet Allamigeon, Xavier
Gaubert, Stéphane
Katz, Ricardo David
author_role author
author2 Gaubert, Stéphane
Katz, Ricardo David
author2_role author
author
dc.subject.none.fl_str_mv MAX-PLUS SEMIRING
MAX-PLUS CONVEXITY
TROPICAL CONVEXITY
POLYHEDRA
HYPERGRAPH TRANSVERSALS
MINIMAL HITTING SETS
MINIMAL SOLUTIONS
topic MAX-PLUS SEMIRING
MAX-PLUS CONVEXITY
TROPICAL CONVEXITY
POLYHEDRA
HYPERGRAPH TRANSVERSALS
MINIMAL HITTING SETS
MINIMAL SOLUTIONS
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a tropical analogue of Farkas lemma, which allows one to check whether a linear inequality is implied by a finite family of linear inequalities. Here, the certificate is a strategy of a mean payoff game. We discuss examples, showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded, and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone.
Fil: Allamigeon, Xavier. Institut National de Recherche en Informatique et en Automatique; Francia
Fil: Gaubert, Stéphane. École Polytechnique; Francia
Fil: Katz, Ricardo David. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura. Instituto de Matemática "Beppo Levi"; Argentina
description We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a tropical analogue of Farkas lemma, which allows one to check whether a linear inequality is implied by a finite family of linear inequalities. Here, the certificate is a strategy of a mean payoff game. We discuss examples, showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded, and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone.
publishDate 2011
dc.date.none.fl_str_mv 2011-10
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/11336/268274
Allamigeon, Xavier; Gaubert, Stéphane; Katz, Ricardo David; Tropical polar cones, hypergraph transversals, and mean payoff games; Elsevier Science Inc.; Linear Algebra and its Applications; 435; 7; 10-2011; 1549-1574
0024-3795
CONICET Digital
CONICET
url http://hdl.handle.net/11336/268274
identifier_str_mv Allamigeon, Xavier; Gaubert, Stéphane; Katz, Ricardo David; Tropical polar cones, hypergraph transversals, and mean payoff games; Elsevier Science Inc.; Linear Algebra and its Applications; 435; 7; 10-2011; 1549-1574
0024-3795
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0024379511001042
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.laa.2011.02.004
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 Elsevier Science Inc.
publisher.none.fl_str_mv Elsevier Science Inc.
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
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score 13.070432

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