A Fast Exact Algorithm for the Optimum Cooperation Problem

Given a graph G=(V,E) with real edge weights, the optimum cooperation problem consists in determining a partition of the graph that maximizes the sum of weights of the edges having nodes in the same partition plus the number of resulting partitions. The problem is also known in the literature as the optimum attack problem in networks. It occurs as a subproblem in the separation of partition inequalities. Furthermore, a relevant physics application exists. Solution algorithms known in the literature require at least |V|-1 minimum cut computations in a corresponding network. In this work, we present a fast exact algorithm for the optimum cooperation problem. By graph-theoretic considerations and appropriately designed heuristics, we considerably reduce the number of minimum cut computations that are necessary in practice. We show the effectiveness of our method by comparing the performance of our algorithm with that of the fastest previously known method on instances coming from the physics application

A Fast Exact Algorithm for the Optimum Cooperation Problem

Given a graph G=(V,E) with real edge weights, the optimum cooperation problem consists in determining a partition of the graph that maximizes the sum of weights of the edges having nodes in the same partition plus the number of resulting partitions. The problem is also known in the literature as the optimum attack problem in networks. It occurs as a subproblem in the separation of partition inequalities. Furthermore, a relevant physics application exists. Solution algorithms known in the literature require at least |V|-1 minimum cut computations in a corresponding network. In this work, we present a fast exact algorithm for the optimum cooperation problem. By graph-theoretic considerations and appropriately designed heuristics, we considerably reduce the number of minimum cut computations that are necessary in practice. We show the effectiveness of our method by comparing the performance of our algorithm with that of the fastest previously known method on instances coming from the physics application

A Fast Exact Algorithm for the Problem of Optimum Cooperation and Structure of Its Solutions

Given a graph with real edge weights, the optimum cooperation problem consists in determining a partition of the graph that maximizes the sum of weights of the edges with nodes in the same class plus the number of the classes of the partition. The problem is also known in the literature as the optimum attack problem in networks. Furthermore, a relevant physics application exists. In this work, we present a fast exact algorithm for the optimum cooperation problem. Algorithms known in the literature require n-1 minimum cut computations in a corresponding network, where n is the number of nodes in the graph. By theoretical considerations and appropriately designed heuristics, we considerably reduce the numbers of minimum cut computations that are necessary in practice. We show the effectiveness of our method by presenting results on instances coming from the physics application. Furthermore, we analyze the structure of the optimal solutions

Regenerator placement in optical networks

Cataloged from PDF version of article.Increase in the number of users and resources consumed by modern applications results in an explosive growth in the traffic on the Internet. Optical networks with higher bandwidths offer faster and more reliable transmission of data and allows transmission of more data. Fiber optical cables have these advantages over the traditional copper wires. So it is expected that optical networks will have a wide application area. However, there are some physical impairments and optical layer constraints in optical networks. One of these is signal degradation which limits the range of optical signals. Signals are degraded during transmission and below a threshold the signals become useless. In order to prevent this, regenerators which are capable of re-amplifying optical signals are used. Since regeneration is a costly process, it is important to decrease the number of regenerators used in an optical network. To increase the reliability of the network, two edge-disjoint paths between each terminal on the network are to be constructed. So the second path could be used in case of a failure in transmitting data on an edge of the first path. Considering these requirements, selecting the nodes on which regenerators are to be placed is an important decision. In this thesis, we discuss the problem of placing signal regenerators on optical networks with restoration. An integer linear program is formulated for this problem. Due to the huge size and other problems of the formulation, it is impractical to use it on large networks. For this reason, a fast heuristic algorithm is proposed to solve this problem. Three methods are proposed to check the feasibility when a fixed set of regenerators are placed on specific nodes. Additionally, a branch and bound algorithm which employs the proposed heuristic is developed to find the optimal solution of our problem. Performance of both the heuristics and the branch and bound method are evaluated in terms of number of regenerators placed and solution times of the algorithms.Özkök, OnurM.S

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

Hub & regenerator location and survivable network design

Ankara : The Department of Industrial Engineering and the Institute of Engineering and Science of Bilkent University, 2010.Thesis (Ph. D.) -- Bilkent University, 2010.Includes bibliographical references leaves 180-184.With the vast development of the Internet, telecommunication networks are employed in numerous different outlets. In addition to voice transmission, which is a traditional utilization, telecommunication networks are now used for transmission of different types of data. As the amount of data transmitted through the network increases, issues such as the survivability and the capacity of the network become more imperative. In this dissertation, we deal with both design and routing problems in telecommunications networks. Our first problem is a two level survivable network design problem. The topmost layer of this network consists of a backbone component where the access equipments that enable the communication of the local access networks are interconnected. The second layer connects the users on the local access network to the access equipments, and consequently to the backbone network. To achieve a survivable network, one that stays operational even under minor breakdowns, the backbone network is assumed to be 2-edge connected while local access networks are to have the star connectivity. Within the literature, such a network is referred to as a 2-edge connected/star network. Since the survivability requirements of networks may change based on the purposes they are utilized for, a variation of this problem in which local access networks are also required to be survivable is also analyzed. The survivability of the local access networks is ensured by providing two connections for every component of the local access networks to the backbone network. This architecture is known as dual homing in the literature. In this dissertation, the polyhedral analysis of the two versions of the two level survivable network design problem is presented; separation problems are analyzed; and branch-and-cut algorithms are developed to find exact solutions. The increased traffic on the telecommunications networks requires the use of high capacity components. Optical networks, composed of fiber optical cables, offer solutions with their higher bandwidths and higher transmission speeds. This makes the optical networks a good alternative to handle the rapid increase in the data traffic. However, due to signal degradation which makes signal regeneration necessary introduces the regenerator placement problem as signal regeneration is a costly process in optical networks. In the regenerator placement problem, we study a location and routing problem together on the backbone component of a given telecommunications network. Survivability is also considered in this problem simultaneously. Exact solution methodologies are developed for this problem: mathematical models and some valid inequalities are proposed; separation problems for the valid inequalities are analyzed and a branch-and-cut algorithm is devised.Özkök, OnurPh.D

On the separation of partition inequalities

Given a graph ¢¤£¦¥¨§�©� � with nonnegative ����� weights for each � edge, a partition inequality is of the ������� form. ��� Here denotes the multicut defined � by of. Partition inequalities arise as valid inequalities for optimization problems related � to-connectivity. In this paper, we will show that, if ¢ decomposes into by 1 and 2-node cutsets, then the separation problem for the partition inequalities on ¢ can be solved by mean of a polynomial time combinatorial algorithm provided that such an algorithm exists for ¢���§������� § ¢ � where ¢���§������� § ¢ � are graphs related to ¢���§�������§� ¢ �. W

On some algorithmic aspects of hypergraphic matroids

International audienceHypergraphics matroids were studied first by Lorea [18] and later by Frank et al [8]. They can be seen as generalizations of graphic matroids. Here we show that several algorithms developed for the graphic case can be extended to hypergraphic matroids. We treat the following: the separation problem for the associated polytope, testing independence, separation of partition inequalities, computing the rank of a set, computing the strength, computing the arboricity and network reinforcement