A Lagrangean Heuristic For The Degree Constrained Minimal Spanning Tree Problem

In this paper we present a new heuristic procedure to solve the degree constrained minimal spanning tree problem. This procedure uses Lagrangian relaxation of the integer programming formulation of the problem to get a lower bound for the optimal objective function value. A subgradient optimization method is used to find multipliers that give good lower bounds. A branch exchange procedure is used after each iteration of the subgradient optimization to generate a feasible solution from an infeasible Lagrangean solution. Computational results are given for problems with up to 300 nodes. The heuristic procedure presented here gives optimal solutions in most instances. For problem sets that were not solved optimally, the gap between the lower bound and the feasible solution was less than 10-2 percent

Exact arborescences, matchings and cycles

AbstractSuppose we are given a graph in which edge has an integral weight. An ‘exact’ problem is to determine whether a desired structure exists for which the sum of the edge weights is exactly k for some prescribed k.We consider the special case of the problem in which all costs are zero or one for arborescences and show that a ‘continuity’ property is prossessed similar to that possessed by matroids. This enables us to determine in polynomial time the complete set of values of k for which a solution exists. We also give a minmax theorem for the maximum possible value of k, in terms of a packing of certain directed cuts in the graph.We also show how enumerative techniques can be used to solve the general exact problem for arborescences (implying spanning trees), perfect matchings in planar graphs and sets of disjoint cycles in a class of planar directed graphs which includes those of degree three. For these problems, we thereby obtain polynomial algorithms provided that the weights are bounded by a constant or encoded in unary

Problema da árvore de suporte de custo mínimo com restrição de grau e custos associados aos nodos

Tese de doutoramento, Estatística e Investigação Operacional (Optimização), 2009, Universidade de Lisboa, Faculdade de CiênciasNesta dissertação aborda-se uma variante de Problemas de Árvores em Grafos onde, para além de custos associados às ligações entre os nodos, existem também custos associados ao grau dos nodos. Esta variante é motivada no contexto de redes de telecomunicações, onde estes custos se encontram associados a equipamento de routing que é necessário instalar em todo os nodos que estejam ligados a mais do que um nodo na rede. Neste tipo de redes é ainda usual restringir o número máximo de ligações de cada vértice de forma a reduzir interferências de sinal. Esta variante é aplicada a dois problemas clássicos de Árvores em Grafos: o problema da Árvore de Suporte de Custo Mínimo e o problema da Árvore de Steiner com Recolha de Prémios. Incorporando esta variante nas formulações tradicionalmente utilizadas para estes problemas chega-se a modelos não lineares, devido à presença dos custos associados ao grau dos vértices. Duas técnicas de reformulação de modelos s aoent ao utilizadas: a técnica de Reformulação por Discretização e a técnica de Reformulação por Caminhos. Ao utilizar qualquer uma destas duas técnicas no modelos tradicionais, obt em-se modelos lineares. Além disso, as duas técnicas permitem construir conjuntos de desigualdades válidas que ao ser adicionadas a um modelo fortalecem-no no que diz respeito à respectiva relaxação linear. A segunda técnica só pode ser aplicada ao primeiro problema devido à existência de uma estrutura de Saco Mochila, presente neste problema. Os modelos apresentados são comparados utilizando um conjunto de instâncias com 25 e 50 nodos.In this dissertation a new variant of Tree Problems in Graphs is considered.In this variant, besides the costs associated to the links between the nodes,there also exist costs associated with the degree of the nodes. This variantis motivated in the context of telecommunications networks where this typeof costs is associated with routing equipment that has to be installed inevery node that is connected with more than one node in the network. Inthis kind of networks it is usual to limit the number of links in any node toprevent signal interferences. This variant is applied to two classical problems:the Spanning Tree Problem and the Prize-collecting Steiner Tree Problem.By integrating this variant in traditional formulations for these problem oneobtains non-linear models, due to the presence of the costs associated withthe degree of the nodes. Two reformulations techniques are then used: theReformulation by Discretization technique and the Reformulation by Pathstechnique. Linear models are obtained by using any of these two techniquesin the traditional models. In addition, different sets of valid inequalities canbe constructed with the use of these techniques which, when added to amodel, strengthen it in terms of the respective linear relaxation. The secondtechnique can only be applied to the first problem due to the existence of aKnapsack structure present in this problem.The presented models are compared using a set of instances with 25 and 50nodes