An O (|E|)-linear model for the maxcut problem
Autor(a) principal: | |
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Data de Publicação: | 2019 |
Tipo de documento: | Tese |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UFPE |
Texto Completo: | https://repositorio.ufpe.br/handle/123456789/33482 |
Resumo: | A polytope P is a model for a combinatorial problem on finite graphs G whose variables are indexed by the edge set E of G if the points of P with (0,1)-coordinates are precisely the characteristic vectors of the subset of edges inducing the feasible configurations for the problem. In the case of the (simple) Max Cut problem, which is the one that concern us here, the feasible subsets of edges are the ones inducing the bipartite subgraphs of G. This work we introduce a new polytope P₁₂ _ R|E given by at most 11|E| inequalities, which is a model for the Max Cut problem on G. Moreover, the left side of each inequality is the sum of at most 4 edge variables with coefficients ±1 and right side 0, 1 or 2. We restrict our analysis to the case of G = Kz, the complete graph in z vertices, where z is an even positive integer z _ 4. This case is sufficient to study because the simple Max Cut problem for general graphs G can be reduced to the complete graph| K z by considering the objective function of the associated integer programming as the characteristic vector of the edges in G _ Kz. This is a polynomial algorithmic transformation. An extension to the linear model into a more complete symmetric model which contains all the permutations for triangular and quadrilateral inequalities, equivalent to other formulations present in the literature is presented as well as the 01-cliques. |
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HENRIQUES, Diogo Brandão Borboremahttp://lattes.cnpq.br/1791617101007702http://lattes.cnpq.br/1018418114348974LINS, Sóstenes Luiz Soares2019-09-23T18:05:48Z2019-09-23T18:05:48Z2019-01-22https://repositorio.ufpe.br/handle/123456789/33482A polytope P is a model for a combinatorial problem on finite graphs G whose variables are indexed by the edge set E of G if the points of P with (0,1)-coordinates are precisely the characteristic vectors of the subset of edges inducing the feasible configurations for the problem. In the case of the (simple) Max Cut problem, which is the one that concern us here, the feasible subsets of edges are the ones inducing the bipartite subgraphs of G. This work we introduce a new polytope P₁₂ _ R|E given by at most 11|E| inequalities, which is a model for the Max Cut problem on G. Moreover, the left side of each inequality is the sum of at most 4 edge variables with coefficients ±1 and right side 0, 1 or 2. We restrict our analysis to the case of G = Kz, the complete graph in z vertices, where z is an even positive integer z _ 4. This case is sufficient to study because the simple Max Cut problem for general graphs G can be reduced to the complete graph| K z by considering the objective function of the associated integer programming as the characteristic vector of the edges in G _ Kz. This is a polynomial algorithmic transformation. An extension to the linear model into a more complete symmetric model which contains all the permutations for triangular and quadrilateral inequalities, equivalent to other formulations present in the literature is presented as well as the 01-cliques.FACEPEUm politopo P é um modelo para um problema combinatorial em um grafo finito G cujas variáveis são indexadas pelo conjunto de arestas E de G se os pontos de P com coordenadas (0,1) são precisamente o vetor característico do subconjunto de arestas induzindo um configuração viável do problema. No caso do Corte Máximo simples, que é o problema abordado neste trabalho, o subconjunto de arestas viáveis é aquele que induz uma bipartição dos vértices de G. Neste trabalho é apresentado um novo politopo P ₁₂ _ R|E| contendo no máximo 11|E| desigualdades, que é um modelo para o problema do Corte Máximo em G. O lado esquerdo de cada inequação é a soma de no máximo quatro variáveis de aresta com coeficientes ±1 e o lado direito é 0, 1 ou 2. A análise é restrita para o caso G = Kz, o grafo completo com z vértices, onde z é um inteiro positivo com z _ 4. Este caso é suficiente pois o problema do Corte Máximo simples para grafos gerais G pode ser reduzido ao grafo completo Kz.engUniversidade Federal de PernambucoPrograma de Pos Graduacao em Ciencia da ComputacaoUFPEBrasilAttribution-NonCommercial-NoDerivs 3.0 Brazilhttp://creativecommons.org/licenses/by-nc-nd/3.0/br/info:eu-repo/semantics/openAccessCiência da computaçãoOtimização combinatóriaAn O (|E|)-linear model for the maxcut probleminfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisdoutoradoreponame:Repositório Institucional da UFPEinstname:Universidade Federal de Pernambuco (UFPE)instacron:UFPETHUMBNAILTESE Diego Brandão Borborema Henriques.pdf.jpgTESE Diego Brandão Borborema Henriques.pdf.jpgGenerated Thumbnailimage/jpeg1316https://repositorio.ufpe.br/bitstream/123456789/33482/5/TESE%20Diego%20Brand%c3%a3o%20Borborema%20Henriques.pdf.jpg86019ef110fb72aba9a8acea270b8662MD55ORIGINALTESE Diego Brandão Borborema Henriques.pdfTESE Diego Brandão Borborema Henriques.pdfapplication/pdf5916995https://repositorio.ufpe.br/bitstream/123456789/33482/1/TESE%20Diego%20Brand%c3%a3o%20Borborema%20Henriques.pdf45ddc9f83bf0a4f8437dadc5ee9f2a1bMD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; 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dc.title.pt_BR.fl_str_mv |
An O (|E|)-linear model for the maxcut problem |
title |
An O (|E|)-linear model for the maxcut problem |
spellingShingle |
An O (|E|)-linear model for the maxcut problem HENRIQUES, Diogo Brandão Borborema Ciência da computação Otimização combinatória |
title_short |
An O (|E|)-linear model for the maxcut problem |
title_full |
An O (|E|)-linear model for the maxcut problem |
title_fullStr |
An O (|E|)-linear model for the maxcut problem |
title_full_unstemmed |
An O (|E|)-linear model for the maxcut problem |
title_sort |
An O (|E|)-linear model for the maxcut problem |
author |
HENRIQUES, Diogo Brandão Borborema |
author_facet |
HENRIQUES, Diogo Brandão Borborema |
author_role |
author |
dc.contributor.authorLattes.pt_BR.fl_str_mv |
http://lattes.cnpq.br/1791617101007702 |
dc.contributor.advisorLattes.pt_BR.fl_str_mv |
http://lattes.cnpq.br/1018418114348974 |
dc.contributor.author.fl_str_mv |
HENRIQUES, Diogo Brandão Borborema |
dc.contributor.advisor1.fl_str_mv |
LINS, Sóstenes Luiz Soares |
contributor_str_mv |
LINS, Sóstenes Luiz Soares |
dc.subject.por.fl_str_mv |
Ciência da computação Otimização combinatória |
topic |
Ciência da computação Otimização combinatória |
description |
A polytope P is a model for a combinatorial problem on finite graphs G whose variables are indexed by the edge set E of G if the points of P with (0,1)-coordinates are precisely the characteristic vectors of the subset of edges inducing the feasible configurations for the problem. In the case of the (simple) Max Cut problem, which is the one that concern us here, the feasible subsets of edges are the ones inducing the bipartite subgraphs of G. This work we introduce a new polytope P₁₂ _ R|E given by at most 11|E| inequalities, which is a model for the Max Cut problem on G. Moreover, the left side of each inequality is the sum of at most 4 edge variables with coefficients ±1 and right side 0, 1 or 2. We restrict our analysis to the case of G = Kz, the complete graph in z vertices, where z is an even positive integer z _ 4. This case is sufficient to study because the simple Max Cut problem for general graphs G can be reduced to the complete graph| K z by considering the objective function of the associated integer programming as the characteristic vector of the edges in G _ Kz. This is a polynomial algorithmic transformation. An extension to the linear model into a more complete symmetric model which contains all the permutations for triangular and quadrilateral inequalities, equivalent to other formulations present in the literature is presented as well as the 01-cliques. |
publishDate |
2019 |
dc.date.accessioned.fl_str_mv |
2019-09-23T18:05:48Z |
dc.date.available.fl_str_mv |
2019-09-23T18:05:48Z |
dc.date.issued.fl_str_mv |
2019-01-22 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
format |
doctoralThesis |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://repositorio.ufpe.br/handle/123456789/33482 |
url |
https://repositorio.ufpe.br/handle/123456789/33482 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.rights.driver.fl_str_mv |
Attribution-NonCommercial-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nc-nd/3.0/br/ info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
Attribution-NonCommercial-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nc-nd/3.0/br/ |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Universidade Federal de Pernambuco |
dc.publisher.program.fl_str_mv |
Programa de Pos Graduacao em Ciencia da Computacao |
dc.publisher.initials.fl_str_mv |
UFPE |
dc.publisher.country.fl_str_mv |
Brasil |
publisher.none.fl_str_mv |
Universidade Federal de Pernambuco |
dc.source.none.fl_str_mv |
reponame:Repositório Institucional da UFPE instname:Universidade Federal de Pernambuco (UFPE) instacron:UFPE |
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UFPE |
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UFPE |
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Repositório Institucional da UFPE |
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Repositório Institucional da UFPE |
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