Modelling and solving complex combinatorial optimization problems : quorumcast routing, elementary shortest path, elementary longest path and agricultural land allocation

The feasible solution set of a Combinatorial Optimization Problem (COP) is discrete and finite. Solving a COP is to find optimal solutions in the set of feasible solutions such that the value of a given cost function is minimized or maximized. In the literature, there exist both complete and incomplete methods for solving COPs. The complete (or exact) methods return the optimal solutions with the proof of the optimality, for example the branch-and-cut search. The incomplete methods try to find hight-quality solutions which are as close to the optimal solutions as possible, for example local search. In this thesis we focus on solving four distinct COPs: the Quorumcast Routing Problem (QRP), the Elementary Shortest Path Problem on graphs with negative-cost cycles (ESPP), the Elementary Longest Path Problem on graphs with positive-cost cycles (ELPP), and the Agricultural Land Allocation Problem (ALAP). In order to solve these problems with the complete methods, we use the Branch-and-Infer search, the Branch-and-Cut search, and the Branch-and-Price search. We also solve ALAP by the incomplete methods, such as Local Search, Tabu Search, Constraints-Based Local Search that combine with metaheuristics. The experimental evaluations on well-known benchmarks show that all proposed algorithms for all the first three COPs (QRP, ESPP and ELPP) are better than the-state-the art algorithms. Specially, we describe ALAP, formulate it as a combination of three COPs, and propose several complete and incomplete algorithms for these COPs.(FSA - Sciences de l'ingénieur) -- UCL, 201

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

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)

Noname manuscript No. (will be inserted by the editor) Solving the Quorumcast Routing Problem by Constraint Programming

Abstract The quorumcast routing problem is a generalization of multicasting which arises in many distributed applications. It consists of finding a minimum cost tree that spans the source node r and at least q out of m specified nodes on a given undirected weighted graph. This paper proposes a complete and an incomplete approach, both based on the same Constraint Programming (CP) model, but with two different specific search heuristics based on shortest paths. Experimental results show the efficiency of the two proposed approaches. Our complete approach (CP model + complete search) is better than the state of the art complete algorithm and our incomplete approach (CP model + incomplete search) is better than the state of the art incomplete algorithm. Moreover, the proposed complete search is better than the standard First-Fail search in the same CP model

LS(Graph): a constraint-based local search framework for constrained optimum tree and path problems on graphs

Constrained Optimum Tree (COT) and Constrained Optimum Path (COP) are two classes of problems which arise in many real-life applications and are ubiquitous in communication networks, transportations, very large scale integration (VLSI) and distributed systems. Most of these problems are computationally very hard to solve. They have been traditionally approached by dedicated algorithms including heuristics and exact algorithms, which are often hard to extend with side constraints and to apply widely because they depend strongly on the problem structures. Moreover, it is required huge research and programming efforts for solving new problems. In this thesis, we construct a constraint-based local search (CBLS) framework, called LS(Graph), for solving COT/COP applications, bringing the compositionality, reuse, and extensibility at the core of CBLS and CP systems. The modeling contribution is the ability to express compositional models for various COT/COP applications at a high level of abstraction, while cleanly separating the model and the search procedure. The LS(Graph) framework will strengthen the modeling benefits of CBLS. By using LS(Graph), users can quickly develop a local search algorithm for a new problem which gives, in most of cases, an acceptable solution while waiting for experts who do research with huge efforts for dedicated algorithms. Moreover, this solution can be used as the initial solution in more complex and hybrid algorithms. The main technical contribution is a connected neighborhood based on rooted spanning trees. The idea behind is to use rooted spanning tree for representing solutions which are paths and their neighborhoods. This approach enables the genericity of the framework from both modeling and computation standpoints. The constructed framework is applied to some three COT (i.e., the edge-weighted k-cardinality tree problem, the quorumcast routing problem, the problem of minimizing congestions on ethernet networks) and four COP problems (i.e., the resource constrained shortest path problem, the edge-disjoint paths problem, the routing and wavelength assignment with delay side constraint problem, and the routing for network covering problem). Experimental results show the potential benefits of the approach. On the one hand, we show the facility and the genericity of the resolution of the COT/COP applications which can be extended with side constraints. On the other hand, for the quorumcast routing and the edge-disjoint paths problems, we show competitive results in comparing with existing techniques.(FSA 3) -- UCL, 201