Network Flow Models for Designing Diameter-Constrained Minimum Spanning and Steiner Trees

The Diameter-Constrained Minimum Spanning Tree Problem seeks a least cost spanning tree subject to a (diameter) bound imposed on the number of edges in the tree between any node pair. A traditional multicommodity flow model with a commodity for every pair of nodes was unable to solve a 20-node and 100-edge problem after one week of computation. We formulate the problem as a directed tree from a selected central node or a selected central edge. Our model simultaneously finds a central node or a central edge and uses it as the source for the commodities in a directed multicommodity flow model with hop constraints. The new model has been able to solve the 20-node, 100-edge instance to optimality after less than four seconds. We also present model enhancements when the diameter bound is odd (these situations are more difficult). We show that the linear programming relaxation of the best formulations discussed in this paper always give an optimal integer solution for two special, polynomially-solvable cases of the problem. We also examine the Diameter Constrained Minimum Steiner Tree problem. We present computational experience in solving problem instances with up to 100 nodes and 1000 edges. The largest model contains more than 250,000 integer variables and more than 125,000 constraints

Extended formulation for hop constrained distribution network configuration problems

International audienceA distribution network is a system aiming to transfer a certain type of resource from feeders to customers. Feeders are producers of a resource and customers have a certain demand in this resource that must be satisfied. Distribution networks can be represented on graphs and be subject to constraints that limit the number of intermediate nodes between some elements of the network (hop constraints) because of physical constraints. This paper uses layered graphs for hop constrained problems to build extended formulations. Preprocessing techniques are also presented to reduce the size of the layered graphs used. The presented model is studied on the hop-constrained minimum margin problem in an electricity network. This problem consists of designing a connected electricity distribution network, and to assign customers to electricity feeders at a maximum number of hops H so as to maximize the minimum capacity margin over the feeders to avoid an overload for any feeder. Numerical results of our model are compared with those of state-of-the-art solution techniques of the minimum margin problem form Rossi et al. [20]. Variations of the initial problem are also presented, considering losses due to transportation or by replacing hop constraints by distance constraints, a variation arising in the context of multicast transmission in telecommunications

Árvore de suporte de custo mínimo com restrições de salto

Mestrado em MatemáticaNeste trabalho descrevemos um algoritmo Dual Ascendente para o problema da Árvore de Suporte de Custo Mínimo com Restrições de Salto (HMST). O problema HMST modela o desenho de uma rede de telecomunicações centralizada com restrições de salto. Estas restrições estão relacionadas com a performance da rede, uma vez que limitam o número de ligações que podem ser utilizadas para ligar o computador central a qualquer um dos terminais e garantem uma certa qualidade de serviço no que diz respeito a alguns critérios de performance tais como disponibilidade, fiabilidade e tempos de atraso máximo de transmissão. Apresentamos duas formulações de fluxos orientadas já apresentadas para este problema. A primeira obtém-se de uma conhecida formulação de fluxos para o problema da Árvore de Suporte de Custo Mínimo adicionando as restrições de salto e a segunda é uma formulação que usa índices de salto, é mais compacta, e foi obtida por Gouveia utilizando a técnica de redefinição de variáveis de Martin. Como o problema é NP-difícil centrámos a nossa atenção na obtenção de um Algoritmo Dual Ascendente para obter um limite inferior para o valor óptimo deste problema e construímos uma heurística baseada na solução Dual Ascendente que nos permitiu obter um limite superior. A técnica Dual Ascendente consiste, essencialmente, numa forma de resolução do problema dual (ou da relaxação lagrangeana ou da relaxação linear) que tira vantagem da estrutura especial que o problema dual tem. Os resultados computacionais que apresentamos para avaliar a qualidade dos valores obtidos indicam que, apesar do algoritmo Dual Ascendente e da heurística baseada na solução dual ascendente permitirem de uma forma muito rápida obter, respectivamente, um limite inferior e um limite superior para o valor óptimo do problema, estes limites são de fraca qualidade. ABSTRACT: In this thesis we describe a Dual Ascent algorithm to the Hop-Constrained Minimum Spanning Tree Problem (HMST). This problem models the design of centralized telecommunication network with hop constraints. These restrictions are related to the network performance. They limit the number of connections that can be used to link the central computer to any of the terminals and they guarantee a certain quality of service with respect to some performance constraints such as availability, reliability and the maximum transmission delay. We present two direct flow formulations already presented for this problem. The first one is obtained from a known flow formulation for the Minimum Spanning Tree Problem adding hop constraints and the second one is a formulation which uses hop-indexes, is more compact and it was obtained by Gouveia using the variable redefinition technique of Martin. As the problem is NP-hard we focus our attention on obtaining a Dual Ascent Algorithm to get to a lower bound to the optimal value of this problem and we build a heuristic based on the dual ascent solution which made it possible to get an upper bound. The Dual Ascent method consists essentially on a way to solve the dual problem (or from the lagrangean relaxation or from the linear relaxation) which takes advantage from the special structure of the dual problem. The computational results we present to evaluate the quality of the obtained values indicate that, although the algorithm Dual Ascent and the heuristic based on the dual ascent solution allow us to obtain in a very rapid way, respectively, a lower and an upper bound for the optimal value of the problem, these bounds are very poor

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

Mestrado em Matemática e AplicaçõesNesta dissertação é apresentada uma implementação de um algoritmo genético para o problema da Árvore de Suporte de Custo Mínimo com Restrições de Salto. Este é um problema de optimização combinatória NP-Difícil, associado a problemas de desenho de redes de telecomunicações centralizadas. Nestas redes um dispositivo central deve ser ligado a outros dispositivos periféricos, sem exceder um número máximo de ligações intermédias, designado por salto H, de forma a garantir a integridade e qualidade do sinal da ligação. O algoritmo genético implementado considera duas codificações para os cromossomas, a codificação por sequências de Prüfer e a codificação por sequências de arestas. A geração da população consiste em dois métodos, um aleatório e um heurístico que considera a restrição de salto do problema. Os resultados computacionais mostram a performance destes dois métodos de geração da população, assim como a influência de vários parâmetros do algoritmo genético na solução obtida, para as duas codificações em estudo. Os parâmetros considerados são: o número máximo de iterações do algoritmo genético, a dimensão da população, a dimensão de um torneio, o número de torneios, a percentagem de mutação e o número de iterações para renovação da população.In this thesis we present a genetic algorithm implementation for the Hop-Constrained Minimum Spanning Tree problem (HMST). This is an NP-Hard combinatorial optimization problem, associated with centralized telecommunications network design problems. In these networks a central device must be connected to other devices, without exceeding the maximum number of in-between connections, known as hop H, in order to ensure the signals integrity and quality. The implemented genetic algorithm considers two chromosome codings, the Pr ufer coding and the edge-set coding. Two methods are considered for generating the population, a random and an heuristic method which considers the hop constraints of the problem. The computational results show the performance of the algorithms for these two methods, as well as the in uence of the genetic algorithm parameters in the solutions, considering the two chromosome codings. The studied genetic algorithm parameters are: maximum number of iterations for the genetic algorithm, population dimension, tournament dimension, number of tournaments, mutation percentage and the number of iterations for population renewal

Benders Decomposition for the Hop-Constrained Survivable Network Design Problem

Given a graph with nonnegative edge weights and node pairs Q, we study the problem of constructing a minimum weight set of edges so that the induced subgraph contains at least K edge-disjoint paths containing at most L edges between each pair in Q. Using the layered representation introduced by Gouveia [Gouveia, L. 1998. Using variable redefinition for computing lower bounds for minimum spanning and Steiner trees with hop constraints. INFORMS J. Comput. 10(2) 180–188], we present a formulation for the problem valid for any K, L ≥ 1. We use a Benders decomposition method to efficiently handle the large number of variables and constraints. We show that our Benders cuts contain constraints used in previous studies to formulate the problem for L = 2, 3, 4, as well as new inequalities when L ≥ 5. Whereas some recent works on Benders decomposition study the impact of the normalization constraint in the dual subproblem, we focus here on when to generate the Benders cuts. We present a thorough computational study of various branch-and-cut algorithms on a large set of instances including the real-based instances from SNDlib. Our best branch-and-cut algorithm combined with an efficient heuristic is able to solve the instances significantly faster than CPLEX 12 on the extended formulation.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Journal of Telecommunications and Information Technology, 2007, nr 4

kwartalni