The Steiner Problem in Graphs

Maculan, Nelson

doi:10.1016/s0304-0208(08)73236-5

Cited by 63 publications

(59 citation statements)

References 25 publications

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“…Theorem 2: All vectors x and y satisfying (1)(2)(3)(4) and (6)(7)(8)(9)(10)(11) are associated with elementary cycles of G.…”

Section: Type Of Variables (Elementary Cycles) Numbermentioning

confidence: 99%

“…All combinatorial optimization problems that will be treated in this paper are associated with connected subgraphs of G. In the section 2 we present a nonsimultaneous flow formulation based on Claus & Maculan (1983), Beasley (1994), Wong (1984), Guyard (1985), Maculan (1986), Maculan (1987) and Maculan, Arpin & Nguyen (1988) for the Steiner tree problem in graphs and in Claus (1984) and Figueiredo & Maculan (2000) for the Travelling Salesman Problem (TSP) to guarantee that a subgraph has to be connected. We characterize all elementary cycles in a graph in section 3, and a particular model is developed for Hamiltonian cycles.…”

Section: Introduction and Definitionsmentioning

confidence: 99%

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Integer linear models with a polynomial number of variables and constraints for some classical combinatorial optimization problems

2003

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“…Theorem 2: All vectors x and y satisfying (1)(2)(3)(4) and (6)(7)(8)(9)(10)(11) are associated with elementary cycles of G.…”

Section: Type Of Variables (Elementary Cycles) Numbermentioning

confidence: 99%

Section: Introduction and Definitionsmentioning

confidence: 99%

Integer linear models with a polynomial number of variables and constraints for some classical combinatorial optimization problems

2003

Self Cite

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“…O objetivo é determinar uma combinação de arcos com custo total mínimo (custos de utilização mais custos dependentes) que conduza os fluxos dos nós de oferta a todos os nós de demanda possivelmente passando por nós intermediários de transbordo, também conhecidos como nós de Steiner. Este é claramente um problema de otimização da classe NP-árdua (do inglês NP-hard), uma vez que generaliza entre outros o conhecido problema de Steiner em grafos (Maculan, 1987), que é NP-árduo (Garey & Johnson, 1979). De fato, a prova da sua NP-complexidade pode ser facilmente desenvolvida .…”

Section: Figura 1 -Grafo G=(n={ijkl} E={(ij)(ji)(ik)(jk)(unclassified

Algoritmos Para O Problema Não Capacitado De Fluxos Com Custos Fixos Nos Arcos: Uma Comparação Estatística

Cruz¹,

Colosimo²,

Mateus

2001

Pesqui. Oper.

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ResumoEste trabalho tem como propósito a apresentação de resultados de uma comparação empírica entre algoritmos, sendo este um dos assuntos mais recorrentes na área de desenvolvimento de algoritmos. Os algoritmos sob estudo são para resolver um problema de otimização em redes, importante pelas suas aplicações potenciais em sistemas de telefonia e transporte, o problema não capacitado de fluxos com custos fixos nos arcos (NCFCF), uma generalização do clássico problema de Steiner em grafos. Para tal, são utilizadas ferramentas estatísticas conhecidas tais como planejamento de experimentos, análise de variância e intervalos de confiança, mas não comumente empregadas neste tipo de estudo. O problema NCFCF é apresentado em uma modelagem de programação matemática inteira mista, baseada na qual os algoritmos sob consideração são apresentados. Uma descrição do planejamento de experimentos adequado a este tipo de estudo é apresentada e é ilustrado o uso da técnica estatística baseado em cuja análise foi possível classificar os algoritmos sob consideração. Palavras-chave: otimização em redes, problemas de Steiner, ANOVA. AbstractThis paper is concerned about empirical comparisons of algorithms, one of the most recurrent issues in the field of design of algorithms. The problem under consideration is the uncapacitated fixed-charge network flow (UFCNF) problem, a generalization of the classic Steiner problem in graphs. The UFCNF is very important in the practical point of view because of its potential applications for telecommunication and transportation system design. In order to compare the algorithms for the UFCNF problem, we make use of well known and well established statistical tools namely the design of experiments, analysis of variance, and confidence intervals, but rarely applied in such studies. A mixed-integer mathematical programming formulation is presented for the UFCNF problem and the design of experiments suitable for the empirical study is detailed. Finally, the statistical analysis based on which the algorithms could be classified is presented.

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“…Finally, constraints (7) impose the condition that there is positive flow on an edge only if the edge is selected. By the max-flow mincut theorem, the projection of the solution onto the variables s satisfy constraints (2) [24]. The results will thus satisfy the following theorem:…”

Section: Ilp Formulationmentioning

confidence: 93%

“…Finally, edge-weights are given by we ∈ R E . The problem of finding a minimum directed Steiner tree rooted at r has previously been examined with an ILP based on graph cuts [5,24,35]:…”

Section: Ilp Formulationmentioning

confidence: 99%

Efficiently Finding the Most Parsimonious Phylogenetic Tree Via Linear Programming

Sridhar

Lam

Blelloch

et al.

Bioinformatics Research and Applications

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Abstract. Reconstruction of phylogenetic trees is a fundamental problem in computational biology. While excellent heuristic methods are available for many variants of this problem, new advances in phylogeny inference will be required if we are to be able to continue to make effective use of the rapidly growing stores of variation data now being gathered. In this paper, we introduce an integer linear programming formulation to find the most parsimonious phylogenetic tree from a set of binary variation data. The method uses a flow-based formulation that could use exponential numbers of variables and constraints in the worst case. The method has, however, proved extremely efficient in practice on datasets that are well beyond the reach of the available provably efficient methods. The program solves several large mtDNA and Y-chromosome instances within a few seconds, giving provably optimal results in times competitive with fast heuristics than cannot guarantee optimality.

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The Steiner Problem in Graphs

Cited by 63 publications

References 25 publications

Integer linear models with a polynomial number of variables and constraints for some classical combinatorial optimization problems

Integer linear models with a polynomial number of variables and constraints for some classical combinatorial optimization problems

Algoritmos Para O Problema Não Capacitado De Fluxos Com Custos Fixos Nos Arcos: Uma Comparação Estatística

Efficiently Finding the Most Parsimonious Phylogenetic Tree Via Linear Programming

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