Construct, Merge, Solve & Adapt A new general algorithm for combinatorial optimization

[EN]This paper describes a general hybrid metaheuristic for combinatorial optimization labelled Construct,Merge, Solve & Adapt. The proposed algorithm is a specific instantiation of a framework known from theliterature as Generate-And-Solve, which is based on the following general idea. First, generate a reducedsub-instance of the original problem instance, in a way such that a solution to the sub-instance is also asolution to the original problem instance. Second, apply an exact solver to the reduced sub-instance inorder to obtain a (possibly) high quality solution to the original problem instance. And third, make use ofthe results of the exact solver as feedback for the next algorithm iteration. The minimum common stringpartition problem and the minimum covering arborescence problem are chosen as test cases in order todemonstrate the application of the proposed algorithm. The obtained results show that the algorithm iscompetitive with the exact solver for small to medium size problem instances, while it significantlyoutperforms the exact solver for larger problem instancesC. Blum was supported by project TIN2012-37930-02 of the Spanish Government. In addition, support is acknowledged from IKERBASQUE (Basque Foundation for Science). J.A. Lozano was partially supported by the IT609-13 program (Basque Government) and project TIN2013-41272P (Spanish Ministry of Science and Innovation)Peer reviewe

Development of hybrid metaheuristics based on instance reduction for combinatorial optimization problems

113 p.La tesis presentada describe el desarrollo de algoritmos metaheurísticos híbridos, basados en reducción de instancias de problema. Éstos son enfocados en la resolución de problemas de optimización combinatorial. La motivación original de la investigación radicó en lograr, a través de la reducción de instancias de problemas, el uso efectivo de modelos de programación lineal entera (ILP) sobre problemas que dado su tamaño no admiten el uso directo con esta técnica exacta. En este contexto se presenta entre otros desarrollos el framework Construct, Merge, Solve & Adapt (CMSA) para resolución de problemas de optimización combinatorial en general, el cual posteriormente fue adaptado para mejorar el desempeño de otras metaheurísticas sin el uso de modelos ILP. Los algoritmos presentados mostraron resultados que compiten o superan el estado del arte sobre los problemas Minimum Common String Partition (MCSP), Minimum Covering Arborescence (MCA) y Weighted Independent Domination (WID)

Development of hybrid metaheuristics based on instance reduction for combinatorial optimization problems

113 p.La tesis presentada describe el desarrollo de algoritmos metaheurísticos híbridos, basados en reducción de instancias de problema. Éstos son enfocados en la resolución de problemas de optimización combinatorial. La motivación original de la investigación radicó en lograr, a través de la reducción de instancias de problemas, el uso efectivo de modelos de programación lineal entera (ILP) sobre problemas que dado su tamaño no admiten el uso directo con esta técnica exacta. En este contexto se presenta entre otros desarrollos el framework Construct, Merge, Solve & Adapt (CMSA) para resolución de problemas de optimización combinatorial en general, el cual posteriormente fue adaptado para mejorar el desempeño de otras metaheurísticas sin el uso de modelos ILP. Los algoritmos presentados mostraron resultados que compiten o superan el estado del arte sobre los problemas Minimum Common String Partition (MCSP), Minimum Covering Arborescence (MCA) y Weighted Independent Domination (WID)

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

Synthesis of Execution Plans for the QVT Core Language

Model transformation languages (MTLs) are important for Model Driven Engineering as they allow the automation of the engineering design process of hardware and software products, in particular at the preliminary and detailed design phases. However, the theories from compiler optimization have not been reused substantively in the development of MTLs. This makes the challenges associated with the implementation of declarative MTLs harder to overcome, in particular with respect to the synthesis of the execution plan (a representation of the control component of the transformation algorithm). The QVT Core MTL is a declarative language, part of a set of standards proposed by the Object Management Group® in order to support the adoption of Model Driven Engineering (MDE). This research presents how instruction scheduling theories can be used for the synthesis of execution plans, in particular for the QVT Core language. The main contributions are a novel approach for performing data dependence analysis on the QVT Core language and its use for the synthesis of execution plans, and the application of metaheuristics to solve the scheduling problem inherent to the synthesis of execution plans. The research demonstrated the feasibility of applying compiler optimization techniques in the design of MTLs and provides a methodology that can be used to construct effi cient execution plans that result in correct transformations. The performance gains and correctness will help the widespread use of the QVT Core language and encourage the adoption of compiler optimization techniques in the implementation of other MTLs

Exact algorithms for network design problems using graph orientations

Gegenstand dieser Dissertation sind exakte Lösungsverfahren für topologische Netzwerkdesignprobleme. Diese kombinatorischen Optimierungsprobleme tauchen in unterschiedlichen realen Anwendungen auf, wie z.B. in der Telekommunikation und der Energiewirtschaft. Die grundlegende Problemstellung dabei ist die Planung bzw. der Ausbau von Netzwerken, die Kunden durch physikalische Leitungen miteinander verbinden. Im Allgemeinen lassen sich solche Probleme graphentheoretisch wie folgt beschreiben: Gegeben eine Menge von Knoten (Kunden, Straßenkreuzungen, Router u.s.w.), eine Menge von Kanten (potenzielle Verbindungsmöglichkeiten) und eine Kostenfunktion auf den Kanten und/oder Knoten. Zu bestimmen ist eine Teilmenge von Knoten und Kanten, so dass die Kostensumme der gewählten Elemente minimiert wird und dabei Nebenbedingungen wie Zusammenhang, Ausfallsicherheit, Kardinalität o.ä. erfüllt werden. In dieser Dissertation behandeln wir zwei spezielle Klassen von topologischen Netzwerkdesignproblemen, nämlich das k-Cardinality Tree Problem (KCT) und das {0,1,2}-Survivable Netzwerkdesignproblem ({0,1,2}- SND) mit Knotenzusammenhang. Diese Probleme sind im Allgemeinen NP-schwer, d.h. nach derzeitigem Stand der Forschung kann es für solche Probleme keine Algorithmen geben die eine optimale Lösung berechnen und dabei für jede mögliche Instanz eine effiziente (d.h. polynomielle) Laufzeit garantieren. Die oben genannten Probleme lassen sich als ganzzahlige lineare Programme (ILPs) formulieren, d.h. als Systeme aus linearen Ungleichungen, ganzzahligen Variablen und einer linearen Zielfunktion. Solche Modelle lassen sich mit Methoden der sogenannten mathematischen Programmierung lösen. Dass die entsprechenden Lösungsverfahren im Allgemeinen sehr zeitaufwendig sein können, war ein oft genutztes Argument für die Entwicklung von (Meta-)Heuristiken um schnell eine Lösung zu erhalten, wenn auch auf Kosten der Optimalität. In dieser Dissertation zeigen wir, dass es, unter Ausnutzung gewisser graphentheoretischer Eigenschaften der zulässigen Lösungen, durchaus möglich ist große anwendungsnahe Probleminstanzen der von uns betrachteten Probleme beweisbar optimal und praktisch-effizient zu lösen. Basierend auf Orientierungseigenschaften der optimalen Lösungen, formulieren wir neue, beweisbar stärkere ILPs und lösen diese anschließend mit Hilfe maßgeschneiderter Branch-and-Cut Algorithmen. Durch umfangreiche polyedrische Analysen können wir beweisen, dass diese Modelle einerseits formal stärkere Beschreibungen der Lösungsräume liefern als bisher bekannte Modelle und andererseits für Branch-and-Cut Verfahren viele praktische Vorteile besitzen. Im Kontext des {0,1,2}-SND geben wir zum ersten Mal eine Orientierungseigenschaft zweiknotenzusammenhängender Graphen an, die zu einer beweisbar stärkeren ILP-Formulierung führt und lösen damit ein in der Literatur seit langem offenes Problem. Unsere experimentellen Ergebnisse für beide Problemklassen zeigen, dass während noch vor kurzem nur Instanzen mit weniger als 200 Knoten in annehmbarer Zeit berechnet werden konnten unsere Algorithmen das optimale Lösen von Instanzen mit mehreren tausend Knoten erlauben. Insbesondere für das KCT Problem ist unser exaktes Verfahren oft sogar schneller als moderne Metaheuristiken, die i.d.R. keine optimale Lösungen finden.The subject of this thesis are exact solution strategies for topological network design problems. These combinatorial optimization problems arise in various real-world scenarios, as, e.g., in the telecommunication and energy industries. The prime task thereby is to plan or extend networks, physically connecting customers. In general we can describe such problems graph-theoretically as follows: Given a set of nodes (customers, street crossings, routers, etc.), a set of edges (potential connections, e.g., cables), and a cost function on the edges and/or nodes. We ask for a subset of nodes and edges, such that the sum of the costs of the selected elements is minimized while satisfying side-conditions as, e.g., connectivity, reliability, or cardinality. In this thesis we concentrate on two special classes of topological network design problems: the k-cardinality tree problem (KCT) and the f0,1,2g-survivable network design problem (f0,1,2g-SND) with node-connectivity constraints. These problems are in general NP-hard, i.e., according to the current knowledge, it is very unlikely that optimal solutions can be found efficiently (i.e., in polynomial time) for all possible problem instances. The above problems can be formulated as integer linear programs (ILPs), i.e., as systems of linear inequalities, integral variables, and a linear objective function. Such models can be solved using methods of mathematical programming. Generally, the corresponding solutions methods can be very time-consuming. This was often used as an argument for developing (meta-)heuristics to obtain solutions fast, although at the cost of their optimality. However, in this thesis we show that, exploiting certain graph-theoretic properties of the feasible solutions, we are able to solve large real-world problem instances to provable optimality efficiently in practice. Based on orientation properties of optimal solutions we formulate new, provably stronger ILPs and solve them via specially tailored branch-and-cut algorithms. Our extensive polyhedral analyses show that these models give tighter descriptions of the solution spaces and also offer certain algorithmic advantages in practice. In the context of f0,1,2g-SND we are able to present the first orientation property of 2-node-connected graphs which leads to a provably stronger ILP formulation, thereby answering a long standing open research question. Until recently, both our problem classes allowed optimal solutions only for instances with roughly up to 200 nodes. Our experimental results show that our new approaches allow instances with thousands of nodes. Especially for the KCT problem, our exact method is often even faster than state-of-the-art metaheuristics, which usually do not find optimal solutions

Target-based Distributionally Robust Minimum Spanning Tree Problem

Due to its broad applications in practice, the minimum spanning tree problem and its all kinds of variations have been studied extensively during the last decades, for which a host of efficient exact and heuristic algorithms have been proposed. Meanwhile, motivated by realistic applications, the minimum spanning tree problem in stochastic network has attracted considerable attention of researchers, with respect to which stochastic and robust spanning tree models and related algorithms have been continuingly developed. However, all of them would be either too restricted by the types of the edge weight random variables or computationally intractable, especially in large-scale networks. In this paper, we introduce a target-based distributionally robust optimization framework to solve the minimum spanning tree problem in stochastic graphs where the probability distribution function of the edge weight is unknown but some statistical information could be utilized to prevent the optimal solution from being too conservative. We propose two exact algorithms to solve it, based on Benders decomposition framework and a modified classical greedy algorithm of MST problem (Prim algorithm),respectively. Compared with the NP-hard stochastic and robust spanning tree problems,The proposed target-based distributionally robust minimum spanning tree problem enjoys more satisfactory algorithmic aspect and robustness, when faced with uncertainty in input data