Computational results for Constrained Minimum Spanning Trees in Flow Networks

In this work, we address the problem of finding a minimum cost spanning tree on a single source flow network. The tree must span all vertices in the given network and satisfy customer demands at a minimum cost. The total cost is given by the summation of the arc setup costs and of the nonlinear flow routing costs over all used arcs. Furthermore, we restrict the trees of interest by imposing a maximum number of arcs on the longest arc emanating from the single source vertex. We propose a dynamic programming model an solution procedure to solve this problem exactly. Intensive computational experiments were performed using randomly generated test problems and the results obtained are reported. From them we can conclude that the method performance is independent of the type of cost functions considered and improves with the tightness of the constrains.Dynamic programming, network flows, constrained trees, general nonlinear costs

On the characterization of the domination of a diameter-constrained network reliability model

AbstractLet G=(V,E) be a digraph with a distinguished set of terminal vertices K⊆V and a vertex s∈K. We define the s,K-diameter of G as the maximum distance between s and any of the vertices of K. If the arcs fail randomly and independently with known probabilities (vertices are always operational), the diameter-constrained s,K-terminal reliability of G, Rs,K(G,D), is defined as the probability that surviving arcs span a subgraph whose s,K-diameter does not exceed D.The diameter-constrained network reliability is a special case of coherent system models, where the domination invariant has played an important role, both theoretically and for developing algorithms for reliability computation. In this work, we completely characterize the domination of diameter-constrained network models, giving a simple rule for computing its value: if the digraph either has an irrelevant arc, includes a directed cycle or includes a dipath from s to a node in K longer than D, its domination is 0; otherwise, its domination is -1 to the power |E|-|V|+1. In particular this characterization yields the classical source-to-K-terminal reliability domination obtained by Satyanarayana.Based on these theoretical results, we present an algorithm for computing the reliability

Problema da árvore de suporte de custo mínimo com restrições de diâmetro

Mestrado em MatemáticaNesta tese desenvolvem-se alguns métodos heurísticos para o Problema da Árvore de Suporte de Custo Mínimo com Restrições de Diâmetro que é um problema de Optimização Combinatória. Este problema tem aplicação na área das telecomunicações e insere-se no âmbito de problemas do Desenho Topológico de Redes de Telecomunicações. O Problema do Desenho de Redes de Terminais consiste em encontrar a melhor maneira de ligar n terminais em diferentes localizações a um nodo central. A topologia óptima deste problema corresponde a uma árvore de suporte de custo mínimo. No Problema da Árvore de Suporte de Custo Mínimo com Restrições de Diâmetro pretende-se determinar uma árvore de suporte de custo mínimo cujo diâmetro não ultrapasse um determinado valor máximo (D). Esta imposição melhora o desempenho da rede. Apresentam-se três heurísticas greedy que seleccionam iterativamente uma aresta a ser incluída na árvore e que se distinguem apenas na forma como são escolhidos os elementos iniciais (nodo/aresta). Descreve-se uma heurística de trocas locais (ou melhoramento) que efectua algumas trocas de arestas de acordo com uma regra estabelecida. Descrevem-se quatro heurísticas de aproximação que adaptam soluções de outro problema ao problema em questão. Na primeira destas heurísticas eliminam-se arestas da árvore de suporte de custo mínimo e, depois, constrói-se a árvore a partir da subárvore obtida. Na segunda proíbe-se a presença na solução de cada uma das arestas de um dado conjunto. Na terceira heurística exige-se que cada aresta de um dado conjunto esteja na solução e, na última exige-se que cada uma das arestas de um dado conjunto esteja na solução e que um conjunto de arestas não esteja na solução. Apresentam-se resultados computacionais que mostram que as Heurísticas de Aproximação são as que obtêm melhores resultados.In this thesis we present some heuristics methods developed to the Diameter constrained Minimum Spanning Tree problem (DMST), which is a Combinatorial Optimization Problem. This is a telecommunication network design problem and the terminal layout problem consists of finding the best way to link n terminals, at different locations, to a central node. The optimal topology for these problems corresponds to a minimum spanning tree. In the DMST we want to obtain a minimum spanning tree which diameter does not surpass a maximum value (D). The diameter constraint improves the performance of the network. We present three greedy heuristics that iteratively select an edge to be added to the tree and are distinguished in the form how initial elements (a node or an edge) are selected. We describe a local exchanges heuristic where improvements are accomplished with some edges exchanges according to an established rule. We also describe four approximation heuristics that adapt solutions from another problem to this problem. On the first heuristic we start it by eliminating edges from the minimal spanning tree and then we build the new tree from the obtained subtree. On the second heuristic, the presence of each edge of a certain set is forbidden in the solution. On the third heuristic, it is demanded that each edge of a certain set is present in the solution, and on the last heuristic it is demanded that each one of the edges of a certain set is present in the solution and that a set of edges is not in the solution. Our computational experience shows that the best results are achieved with the approximation heuristics