Robust Reoptimization of Steiner Trees

In reoptimization, one is given an optimal solution to a problem instance and a (locally) modified instance. The goal is to obtain a solution for the modified instance. We aim to use information obtained from the given solution in order to obtain a better solution for the new instance than we are able to compute from scratch. In this paper, we consider Steiner tree reoptimization and address the optimality requirement of the provided solution. Instead of assuming that we are provided an optimal solution, we relax the assumption to the more realistic scenario where we are given an approximate solution with an upper bound on its performance guarantee. We show that for Steiner tree reoptimization there is a clear separation between local modifications where optimality is crucial for obtaining improved approximations and those instances where approximate solutions are acceptable starting points. For some of the local modifications that have been considered in previous research, we show that for every fixed ε>0, approximating the reoptimization problem with respect to a given (1+ε)-approximation is as hard as approximating the Steiner tree problem itself. In contrast, with a given optimal solution to the original problem it is known that one can obtain considerably improved results. Furthermore, we provide a new algorithmic technique that, with some further insights, allows us to obtain improved performance guarantees for Steiner tree reoptimization with respect to all remaining local modifications that have been considered in the literature: a required node of degree more than one becomes a Steiner node; a Steiner node becomes a required node; the cost of one edge is increased

Improved Algorithms for the Steiner Problem in Networks

We present several new techniques for dealing with the Steiner problem in (undirected) networks. We consider them as building blocks of an exact algorithm, but each of them could also be of interest in its own right. First, we consider some relaxations of integer programming formulations of this problem and investigate different methods for dealing with these relaxations, not only to obtain lower bounds, but also to get additional information which is used in the computation of upper bounds and in reduction techniques. Then, we modify some known reduction tests and introduce some new ones. We integrate some of these tests into a package with a small worst case time which achieves impressive reductions on a wide range of instances. On the side of upper bounds, we introduce the new concept of heuristic reductions. On the basis of this concept, we develop heuristics that achieve sharper upper bounds than the strongest known heuristics for this problem despite running times which are smaller by orders of magnitude. Finally, we integrate these blocks into an exact algorithm. We present computational results on a variety of benchmark instances. The results are clearly superior to those of all other exact algorithms known to the authors

The capacitated minimum spanning tree problem

In this thesis we focus on the Capacitated Minimum Spanning Tree (CMST), an extension of the minimum spanning tree (MST) which considers a central or root vertex which receives and sends commodities (information, goods, etc) to a group of terminals. Such commodities flow through links which have capacities that limit the total flow they can accommodate. These capacity constraints over the links result of interest because in many applications the capacity limits are inherent. We find the applications of the CMST in the same areas as the applications of the MST; telecommunications network design, facility location planning, and vehicle routing. The CMST arises in telecommunications networks design when the presence of a central server is compulsory and the flow of information is limited by the capacity of either the server or the connection lines. Its study also results specially interesting in the context of the vehicle routing problem, due to the utility that spanning trees can have in constructive methods. By the simple fact of adding capacity constraints to the MST problem we move from a polynomially solvable problem to a non-polynomial one. In the first chapter we describe and define the problem, introduce some notation, and present a review of the existing literature. In such review we include formulations and exact methods as well as the most relevant heuristic approaches. In the second chapter two basic formulations and the most used valid inequalities are presented. In the third chapter we present two new formulations for the CMST which are based on the identification of subroots (vertices directly connected to the root). One way of characterizing CMST solutions is by identifying the subroots and the vertices assigned to them. Both formulations use binary decision variables y to identify the subroots. Additional decision variables x are used to represent the elements (arcs) of the tree. In the second formulation the set of x variables is extended to indicate the depth of the arcs in the tree. For each formulation we present families of valid inequalities and address the separation problem in each case. Also a solution algorithm is proposed. In the fourth chapter we present a biased random-key genetic algorithm (BRKGA) for the CMST. BRKGA is a population-based metaheuristic, that has been used for combinatorial optimization. Decoders, solution representation and exploring strategies are presented and discussed. A final algorithm to obtain upper bounds for the CMST is proposed. Numerical results for the BRKGA and two cutting plane algorithms based on the new formulations are presented in the fifth chapter . The above mentioned results are discussed and analyzed in this same chapter. The conclusion of this thesis are presented in the last chapter, in which we include the opportunity areas suitable for future research.En esta tesis nos enfocamos en el problema del Árbol de Expansión Capacitado de Coste Mínimo (CMST, por sus siglas en inglés), que es una extensión del problema del árbol de expansión de coste mínimo (MST, por sus siglas en inglés). El CMST considera un vértice raíz que funciona como servidor central y que envía y recibe bienes (información, objetos, etc) a un conjunto de vértices llamados terminales. Los bienes solo pueden fluir entre el servidor y las terminales a través de enlaces cuya capacidad es limitada. Dichas restricciones sobre los enlaces dan relevancia al problema, ya que existen muchas aplicaciones en que las restricciones de capacidad son de vital importancia. Dentro de las áreas de aplicación del CMST más importantes se encuentran las relacionadas con el diseño de redes de telecomunicación, el diseño de rutas de vehículos y problemas de localización. Dentro del diseño de redes de telecomunicación, el CMST está presente cuando se considera un servidor central, cuya capacidad de transmisión y envío está limitada por las características de los puertos del servidor o de las líneas de transmisión. Dentro del diseño de rutas de vehículos el CMST resulta relevante debido a la influencia que pueden tener los árboles en el proceso de construcción de soluciones. Por el simple de añadir las restricciones de capacidad, el problema pasa de resolverse de manera exacta en tiempo polinomial usando un algoritmo voraz, a un problema que es muy difícil de resolver de manera exacta. En el primer capítulo se describe y define el problema, se introduce notación y se presenta una revisión bibliográfica de la literatura existente. En dicha revisión bibliográfica se incluyen formulaciones, métodos exactos y los métodos heurísticos utilizados más importantes. En el siguiente capítulo se muestran dos formulaciones binarias existentes, así como las desigualdades válidas más usadas para resolver el CMST. Para cada una de las formulaciones propuestas, se describe un algoritmo de planos de corte. Dos nuevas formulaciones para el CMST se presentan en el tercer capítulo. Dichas formulaciones estás basadas en la identificación de un tipo de vértices especiales llamados subraíces. Los subraíces son aquellos vértices que se encuentran directamente conectados al raíz. Un forma de caracterizar las soluciones del CMST es a través de identificar los nodos subraíces y los nodos dependientes a ellos. Ambas formulaciones utilizan variables para identificar los subraices y variables adicionales para identificar los arcos que forman parte del árbol. Adicionalmente, las variables en la segunda formulación ayudan a identificar la profundidad con respecto al raíz a la que se encuentran dichos arcos. Para cada formulación se presentan desigualdades válidas y se plantean procedimientos para resolver el problema de su separación. En el cuarto capítulo se presenta un algoritmo genético llamado BRKGA para resolver el CMST. El BRKGA está basado en el uso de poblaciones generadas por secuencias de números aleatorios, que posteriormente evolucionan. Diferentes decodificadores, un método de búsqueda local, espacios de búsqueda y estrategias de exploración son presentados y analizados. El capítulo termina presentando un algoritmo final que permite la obtención de cotas superiores para el CMST. Los resultados computacionales para el BRKGA y los dos algoritmos de planos de corte basados en las formulaciones propuestas se muestran en el quinto capítulo. Dichos resultados son analizados y discutidos en dicho capítulo. La tesis termina presentando las conclusiones derivadas del desarrollo del trabajo de investigación, así como las áreas de oportunidad sobre las que es posible realizar futuras investigaciones

Bio-Inspired Computing For Complex And Dynamic Constrained Problems

Bio-inspired algorithms are general-purpose optimisation methods that can find solutions with high qualities for complex problems. They are able to find these solutions with minimal knowledge of a search space. Bio-inspired algorithms (the design of which is inspired by nature) can easily adapt to changing environments. In this thesis, we contribute to the theoretical and empirical understanding of bioinspired algorithms, such as evolutionary algorithms and ant colony optimisation. We address complex problems as well as problems with dynamically changing constraints. Firstly, we review the most recent achievements in the theoretical analysis of dynamic optimisation via bio-inspired algorithms. We then continue our investigations in two major areas: static and dynamic combinatorial problems. To tackle static problems, we study the evolutionary algorithms that are enhanced by using a knowledge-based mutation approach in solving single- and multi-objective minimum spanning tree (MST) problems. Our results show that proper development of biased mutation can significantly improve the performance of evolutionary algorithms. Afterwards, we analyse the ability of single- and multi-objective algorithms to solve the packing while travelling (PWT) problem. This NP-hard problem is chosen to represent real-world multi-component problems. We outline the limitations of randomised local search in solving PWT and prove the advantage of using evolutionary algorithms. Our dynamic investigations begin with an empirical analysis of the ability of simple and advanced evolutionary algorithms to optimise the dynamic knapsack (KP) problem. We show that while optimising a population of solutions can speed up the ability of an algorithm to find optimal solutions after a dynamic change, it has the exact opposite effect in environments with high-frequency changes. Finally, we investigate the dynamic version of a more general problem known as the subset selection problem. We prove the inability of the adaptive greedy approach to maintain quality solutions in dynamic environments and illustrate the advantage of using evolutionary algorithms theoretically and practically.Thesis (Ph.D.) -- University of Adelaide, School of Computer Science, 202