Application of biased random-key genetic algorithm and formulations for the Grundy coloring problem and the connected Grundy coloring problem
Autor(a) principal: | |
---|---|
Data de Publicação: | 2023 |
Tipo de documento: | Dissertação |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UFBA |
Texto Completo: | https://repositorio.ufba.br/handle/ri/39322 |
Resumo: | Given a graph G, its Grundy number Γ(G) defines the worst-case behavior for the wellknown and widely used first-fit greedy coloring heuristic. Specifically, Γ(G) is the largest k for which a k-coloring can be obtained with the first-fit heuristic. The connected Grundy number Γc(G) gives the worst-case behavior for the connected first-fit coloring heuristic, that is, one in which each vertex to be colored, except the first, is added adjacent to an already colored vertex. Both problems are NP-hard. In this master’s thesis, we present heuristic and exact approaches to the Grundy coloring problem and the connected Grundy coloring problem, which are optimization problems consisting of obtaining the Grundy number and the connected Grundy number, respectively. This study proposes the use of a algorithm Biased Random-Key Genetic Algorithm (BRKGA) and the use of integer programming formulations using a more traditional (standard) approach and a representative one. A new combinatorial upper bound is also proposed that is valid for both problems and an algorithm using dynamic programming for its calculation. The computational experiments show that the new upper bound can improve over a well-established combinatorial bound available in the literature for several instances. The results also evidence that the formulation by representatives has an overall superior performance than the standard formulation, achieving better results for the denser instances, while the latter performs better for the sparser ones to the Grundy coloring problem. However, we show that these types of integer programming formulations are computationally impractical for the connected version. Furthermore, the BRKGA can find high-quality solutions for both problems and can be used with confidence in large instances where the formulations fail for the Grundy coloring problem. |
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2024-04-26T21:51:54Z2024-04-262024-04-26T21:51:54Z2023-12-13SILVA, Mateus Carvalho da. Application of biased random-key genetic algorithm and formulations for the Grundy coloring problem and the connected Grundy coloring problem. 2023. 49 f. Dissertação (Mestrado em Ciência da Computação) Instituto de Computação, Universidade Federal da Bahia, Salvador, BA, 2023.https://repositorio.ufba.br/handle/ri/39322Given a graph G, its Grundy number Γ(G) defines the worst-case behavior for the wellknown and widely used first-fit greedy coloring heuristic. Specifically, Γ(G) is the largest k for which a k-coloring can be obtained with the first-fit heuristic. The connected Grundy number Γc(G) gives the worst-case behavior for the connected first-fit coloring heuristic, that is, one in which each vertex to be colored, except the first, is added adjacent to an already colored vertex. Both problems are NP-hard. In this master’s thesis, we present heuristic and exact approaches to the Grundy coloring problem and the connected Grundy coloring problem, which are optimization problems consisting of obtaining the Grundy number and the connected Grundy number, respectively. This study proposes the use of a algorithm Biased Random-Key Genetic Algorithm (BRKGA) and the use of integer programming formulations using a more traditional (standard) approach and a representative one. A new combinatorial upper bound is also proposed that is valid for both problems and an algorithm using dynamic programming for its calculation. The computational experiments show that the new upper bound can improve over a well-established combinatorial bound available in the literature for several instances. The results also evidence that the formulation by representatives has an overall superior performance than the standard formulation, achieving better results for the denser instances, while the latter performs better for the sparser ones to the Grundy coloring problem. However, we show that these types of integer programming formulations are computationally impractical for the connected version. Furthermore, the BRKGA can find high-quality solutions for both problems and can be used with confidence in large instances where the formulations fail for the Grundy coloring problem.Dado um grafo G, seu número de Grundy Γ(G) define o comportamento de pior caso para a conhecida e amplamente utilizada heurística de coloração gulosa first-fit. Mais especificamente, Γ(G) é o maior k para o qual uma k-coloração pode ser obtida com a heurística first-fit. O número de Grundy conexo Γc(G) fornece o comportamento do pior caso para a heurística de coloração first-fit conexa, ou seja, aquela em que cada vértice a ser colorido, exceto o primeiro, é adicionado adjacente a um vértice já colorido. Ambos os problemas são NP-difíceis. Nesta dissertação, apresentamos abordagens heurísticas e exatas para o problema da coloração de Grundy e o problema de coloração de Grundy conexo, que são problemas de otimização consistindo na obtenção do número de Grundy e do número de Grundy conexo, respectivamente. Nesse estudo é proposto o uso do algoritmo genético de chaves aleatórias viesado (Biased random-key genetic algorithm - BRKGA) e do uso de formulações de programação inteira usando uma abordagem mais tradicional (padrão) e uma por representativos. Também é proposto um novo limite superior combinatório que é válido para ambos os problemas e um algoritmo usando programação dinâmica para o seu cálculo. Os experimentos computacionais mostram que o novo limite superior pode melhorar o limite para vários casos em relação a um limite combinatório bem estabelecido disponível na literatura. Os resultados também evidenciam que a formulação por representativos tem um desempenho geral superior que a formulação padrão, alcançando melhores resultados para as instâncias mais densas, enquanto esta última tem melhor desempenho para as mais esparsas para o problema dos números de Grundy. Contudo mostramos que este tipo de formulações com programação inteira são computacionalmente impraticáveis para a versão conexa. Além disso, o BRKGA pode encontrar soluções de alta qualidade para ambos os problemas e pode ser usado com confiança em grandes instâncias onde as formulações falham para o problema da coloração de GrundyCoordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).engUniversidade Federal da BahiaPrograma de Pós-Graduação em Ciência da Computação (PGCOMP) UFBABrasilInstituto de Computação - ICCombinatorial optimizationGraph coloringGrundy numberBRKGAWorst-case analysisCNPQ::CIENCIAS EXATAS E DA TERRA::CIENCIA DA COMPUTACAOOtimização combinatóriaColoração de grafosNúmero de GrundyBRKGAAnálise de pior casoApplication of biased random-key genetic algorithm and formulations for the Grundy coloring problem and the connected Grundy coloring problemAplicação do algoritmo genético de chaves aleatórias viesado e formulações para o problema da coloração de Grundy e o problema da coloração conexa de GrundyMestrado Acadêmicoinfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/publishedVersionMelo, Rafael Augusto de0000-0003-4300-0097http://lattes.cnpq.br/4117373032501782Santos, Márcio Costa0000-0002-5452-0226http://lattes.cnpq.br/4258661430014987Melo, Rafael Augusto0000-0003-4300-0097http://lattes.cnpq.br/4117373032501782Silva, Pedro Henrique GonzálesDurão, Frederico Araújo0000-0002-7766-6666http://lattes.cnpq.br/6271096128174325Resende, Maurício Guilherme de Carvalho0000-0001-7462-6207http://lattes.cnpq.br/6235837317096398http://lattes.cnpq.br/8638009209427420Silva, Mateus Carvalhoinfo:eu-repo/semantics/openAccessreponame:Repositório Institucional da UFBAinstname:Universidade Federal da Bahia (UFBA)instacron:UFBAORIGINALMSC_Mateus_Carvalho_da_Silva.pdfMSC_Mateus_Carvalho_da_Silva.pdfDissertação de Mestrado de Mateus Carvalho da Silva.application/pdf1238360https://repositorio.ufba.br/bitstream/ri/39322/1/MSC_Mateus_Carvalho_da_Silva.pdfa667a52981f7827dffc5855354329bbbMD51open accessLICENSElicense.txtlicense.txttext/plain1720https://repositorio.ufba.br/bitstream/ri/39322/2/license.txtd9b7566281c22d808dbf8f29ff0425c8MD52open accessri/393222024-04-26 18:51:54.49open accessoai:repositorio.ufba.br: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Repositório InstitucionalPUBhttp://192.188.11.11:8080/oai/requestopendoar:19322024-04-26T21:51:54Repositório Institucional da UFBA - Universidade Federal da Bahia (UFBA)false |
dc.title.pt_BR.fl_str_mv |
Application of biased random-key genetic algorithm and formulations for the Grundy coloring problem and the connected Grundy coloring problem |
dc.title.alternative.pt_BR.fl_str_mv |
Aplicação do algoritmo genético de chaves aleatórias viesado e formulações para o problema da coloração de Grundy e o problema da coloração conexa de Grundy |
title |
Application of biased random-key genetic algorithm and formulations for the Grundy coloring problem and the connected Grundy coloring problem |
spellingShingle |
Application of biased random-key genetic algorithm and formulations for the Grundy coloring problem and the connected Grundy coloring problem Silva, Mateus Carvalho CNPQ::CIENCIAS EXATAS E DA TERRA::CIENCIA DA COMPUTACAO Otimização combinatória Coloração de grafos Número de Grundy BRKGA Análise de pior caso Combinatorial optimization Graph coloring Grundy number BRKGA Worst-case analysis |
title_short |
Application of biased random-key genetic algorithm and formulations for the Grundy coloring problem and the connected Grundy coloring problem |
title_full |
Application of biased random-key genetic algorithm and formulations for the Grundy coloring problem and the connected Grundy coloring problem |
title_fullStr |
Application of biased random-key genetic algorithm and formulations for the Grundy coloring problem and the connected Grundy coloring problem |
title_full_unstemmed |
Application of biased random-key genetic algorithm and formulations for the Grundy coloring problem and the connected Grundy coloring problem |
title_sort |
Application of biased random-key genetic algorithm and formulations for the Grundy coloring problem and the connected Grundy coloring problem |
author |
Silva, Mateus Carvalho |
author_facet |
Silva, Mateus Carvalho |
author_role |
author |
dc.contributor.advisor1.fl_str_mv |
Melo, Rafael Augusto de |
dc.contributor.advisor1ID.fl_str_mv |
0000-0003-4300-0097 |
dc.contributor.advisor1Lattes.fl_str_mv |
http://lattes.cnpq.br/4117373032501782 |
dc.contributor.advisor-co1.fl_str_mv |
Santos, Márcio Costa |
dc.contributor.advisor-co1ID.fl_str_mv |
0000-0002-5452-0226 |
dc.contributor.advisor-co1Lattes.fl_str_mv |
http://lattes.cnpq.br/4258661430014987 |
dc.contributor.referee1.fl_str_mv |
Melo, Rafael Augusto |
dc.contributor.referee1ID.fl_str_mv |
0000-0003-4300-0097 |
dc.contributor.referee1Lattes.fl_str_mv |
http://lattes.cnpq.br/4117373032501782 |
dc.contributor.referee2.fl_str_mv |
Silva, Pedro Henrique Gonzáles |
dc.contributor.referee3.fl_str_mv |
Durão, Frederico Araújo |
dc.contributor.referee3ID.fl_str_mv |
0000-0002-7766-6666 |
dc.contributor.referee3Lattes.fl_str_mv |
http://lattes.cnpq.br/6271096128174325 |
dc.contributor.referee4.fl_str_mv |
Resende, Maurício Guilherme de Carvalho |
dc.contributor.referee4ID.fl_str_mv |
0000-0001-7462-6207 |
dc.contributor.referee4Lattes.fl_str_mv |
http://lattes.cnpq.br/6235837317096398 |
dc.contributor.authorLattes.fl_str_mv |
http://lattes.cnpq.br/8638009209427420 |
dc.contributor.author.fl_str_mv |
Silva, Mateus Carvalho |
contributor_str_mv |
Melo, Rafael Augusto de Santos, Márcio Costa Melo, Rafael Augusto Silva, Pedro Henrique Gonzáles Durão, Frederico Araújo Resende, Maurício Guilherme de Carvalho |
dc.subject.cnpq.fl_str_mv |
CNPQ::CIENCIAS EXATAS E DA TERRA::CIENCIA DA COMPUTACAO |
topic |
CNPQ::CIENCIAS EXATAS E DA TERRA::CIENCIA DA COMPUTACAO Otimização combinatória Coloração de grafos Número de Grundy BRKGA Análise de pior caso Combinatorial optimization Graph coloring Grundy number BRKGA Worst-case analysis |
dc.subject.por.fl_str_mv |
Otimização combinatória Coloração de grafos Número de Grundy BRKGA Análise de pior caso |
dc.subject.other.pt_BR.fl_str_mv |
Combinatorial optimization Graph coloring Grundy number BRKGA Worst-case analysis |
description |
Given a graph G, its Grundy number Γ(G) defines the worst-case behavior for the wellknown and widely used first-fit greedy coloring heuristic. Specifically, Γ(G) is the largest k for which a k-coloring can be obtained with the first-fit heuristic. The connected Grundy number Γc(G) gives the worst-case behavior for the connected first-fit coloring heuristic, that is, one in which each vertex to be colored, except the first, is added adjacent to an already colored vertex. Both problems are NP-hard. In this master’s thesis, we present heuristic and exact approaches to the Grundy coloring problem and the connected Grundy coloring problem, which are optimization problems consisting of obtaining the Grundy number and the connected Grundy number, respectively. This study proposes the use of a algorithm Biased Random-Key Genetic Algorithm (BRKGA) and the use of integer programming formulations using a more traditional (standard) approach and a representative one. A new combinatorial upper bound is also proposed that is valid for both problems and an algorithm using dynamic programming for its calculation. The computational experiments show that the new upper bound can improve over a well-established combinatorial bound available in the literature for several instances. The results also evidence that the formulation by representatives has an overall superior performance than the standard formulation, achieving better results for the denser instances, while the latter performs better for the sparser ones to the Grundy coloring problem. However, we show that these types of integer programming formulations are computationally impractical for the connected version. Furthermore, the BRKGA can find high-quality solutions for both problems and can be used with confidence in large instances where the formulations fail for the Grundy coloring problem. |
publishDate |
2023 |
dc.date.issued.fl_str_mv |
2023-12-13 |
dc.date.accessioned.fl_str_mv |
2024-04-26T21:51:54Z |
dc.date.available.fl_str_mv |
2024-04-26 2024-04-26T21:51:54Z |
dc.type.driver.fl_str_mv |
Mestrado Acadêmico info:eu-repo/semantics/masterThesis |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
format |
masterThesis |
status_str |
publishedVersion |
dc.identifier.citation.fl_str_mv |
SILVA, Mateus Carvalho da. Application of biased random-key genetic algorithm and formulations for the Grundy coloring problem and the connected Grundy coloring problem. 2023. 49 f. Dissertação (Mestrado em Ciência da Computação) Instituto de Computação, Universidade Federal da Bahia, Salvador, BA, 2023. |
dc.identifier.uri.fl_str_mv |
https://repositorio.ufba.br/handle/ri/39322 |
identifier_str_mv |
SILVA, Mateus Carvalho da. Application of biased random-key genetic algorithm and formulations for the Grundy coloring problem and the connected Grundy coloring problem. 2023. 49 f. Dissertação (Mestrado em Ciência da Computação) Instituto de Computação, Universidade Federal da Bahia, Salvador, BA, 2023. |
url |
https://repositorio.ufba.br/handle/ri/39322 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Universidade Federal da Bahia |
dc.publisher.program.fl_str_mv |
Programa de Pós-Graduação em Ciência da Computação (PGCOMP) |
dc.publisher.initials.fl_str_mv |
UFBA |
dc.publisher.country.fl_str_mv |
Brasil |
dc.publisher.department.fl_str_mv |
Instituto de Computação - IC |
publisher.none.fl_str_mv |
Universidade Federal da Bahia |
dc.source.none.fl_str_mv |
reponame:Repositório Institucional da UFBA instname:Universidade Federal da Bahia (UFBA) instacron:UFBA |
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