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We consider the design of transparent optical networks from a practical perspective. Network operators aim at satisfying the communication demands at minimum cost. Such an optimization involves three interdependent planning issues: the dimensioning of the physical topology, the routing of lightpaths, and the wavelength assignment. Further topics include the reliability of the configuration and sparse wavelength conversion for efficient use of the capacities. In this paper, we investigate this extensive optical network design task. Using a flexible device-based model, we present an integer programming formulation that supports greenfield planning as well as expansion planning on top of an existing network. As solution method, we propose a suitable decomposition approach that separates the wavelength assignment from the dimensioning and routing. Our method in particular provides a lower bound on the total cost which allows to rate the solution quality. Computational experiments on realistic networks approve the solution approach to be appropriate.
In this paper, we study the minimum converter wavelength assignment problem in optical networks. To benchmark the quality of solutions obtained by heuristics, we derive an integer programming formula tion by generalizing the formulation of Mehrotra and Trick (1996) for the vertex coloring problem. To handle the exponential number of variables, we propose a column generation approach. Computational experiments show that the value of the linear relaxation states a good lower bound and can often prove optimality of the best solution generated heuristically.
Linear Programming Lower Bounds for Minimum Converter Wavelength Assignment in Optical Networks
(2004)
In this paper, we study the conflict-free assignment of wavelengths to lightpaths in an optical network with the opportunity to place wavelength converters. To benchmark heuristics for the problem, we develop integer programming formulations and study their properties. Moreover, we study the computational performance of the column generation algorithm for solving the linear relaxation of the most promising formulation. In many cases, a non-zero lower bound on the number of required converters is generated this way. For several instances, we in fact prove optimality since the lower bound equals the best known solution value.
Der scharfe Wettbewerb innerhalb der Telekommunikationsbranche zwingt die Netzbetreiber dazu, ihre Investitionen genau zu planen und immer wieder Einsparungsmanahmen durchzuführen. Gleichzeitig ist es jedoch wichtig, die Qualität der angebotenen Dienste zu verbessern, um neue Kunden zu gewinnen und langfristig an sich zu binden. Die mathematische Optimierung bietet sich für viele solcher Aufgabenstellungen als hervorragend geeignetes Planungswerkzeug an. Ziel dieses Artikels ist es, ihre Methodik und ihre Anwendung speziell zur Kosten- und Qualitätsoptimierung in Kommunikationsnetzen vorzustellen. Anhand von vier konkreten Planungsaufgaben aus dem Bereich der Festnetzplanung wird aufgezeigt, wie sich komplexe Zusammenhänge in flexiblen mathematischen Modellen abbilden lassen und welche Verfahren zur automatisierten Bearbeitung der Probleme eingesetzt werden können. Die hier vorgestellten Methoden zeichnen sich insbesondere dadurch aus, dass sie neben hochwertigen Lösungen auch eine Qualittsgarantie liefern, mit der sich die Lsungen fundiert bewerten lassen. Die dokumentierten Ergebnisse aus verschiedenen Industrieprojekten belegen die Eignung und Güte der mathematischen Optimierung für die Praxis.
Beim Entwurf und Ausbau von Informations- und Kommunikationsnetzwerken m{ü}ssen zahlreiche interdependente Entscheidungen getroffen und gleichzeitig mannigfaltige Bedingungen ber{ü}cksichtigt werden. Die verf{ü}gbaren technischen und organisatorischen Alternativm{ö}glichkeiten sind normalerweise so vielf{ä}ltig und komplex, dass eine manuelle Planung praktisch nicht m{ö}glich ist. In diesem Artikel wird das Potential und die Methodik der mathematischen Optimierung bei der kostenoptimalen Planung von Kommunikationsnetzen vorgestellt. Als Ausgangspunkt wird exemplarisch eine typische praktische Aufgabe, die Struktur- und Konfigurationsplanung mehrstufiger Telekommunikationsnetzwerke, dargestellt. Anschließend werden kurz die wesentlichen Modellierungstechniken und Verfahrensans{ä}tze der mathematischen Optimierung skizziert. Abschließend gehen wir auf die Planung einer ad{ä}quaten Informations- und Kommunikations- Infrastruktur f{ü}r ein dezentrales Energieversorgungsnetz ein.
Stable multi-sets are an evident generalization of the well-known stable sets. As integer programs, they constitute a general structure which allows for a wide applicability of the results. Moreover, the study of stable multi-sets provides new insights to well-known properties of stable sets. In this paper, we continue our investigations started in [{\sl Koster and Zymolka 2002}] and present results of three types: on the relation to other combinatorial problems, on the polyhedral structure of the stable multi-set polytope, and on the computational impact of the polyhedral results. First of all, we embed stable multi-sets in a framework of generalized set packing problems and point out several relations. The second part discusses properties of the stable multi-set polytope. We show that the vertices of the linear relaxation are half integer and have a special structure. Moreover, we strengthen the conditions for cycle inequalities to be facet defining, show that the separation problem for these inequalities is polynomial time solvable, and discuss the impact of chords in cycles. The last result allows to interpret cliques as cycles with many chords. The paper is completed with a computational study to the practical importance of the cycle inequalities. The computations show that the performance of state-of-the-art integer programming solvers can be improved significantly by including these inequalities.
In this paper, a new shared protection mechanism for meshed optical networks is presented. Significant network design cost reductions can be achieved in comparison to the well-known 1+1 protection scheme. Demand-wise Shared Protection (DSP) bases on the diversification of demand routings and exploits the network connectivity to restrict the number of backup lightpaths needed to provide the desired level of prorection. Computational experiments approve the benefits of the concept DSP for cost efficient optical network designs.
Finding conflict-free wavelength assignments with a minimum number of required conversions for a routing of the lightpaths is one of the important tasks within the design of all-optical networks. We consider this problem in multi-fiber networks with different types of WDM systems. We give a detailed description of the problem and derive its theoretical complexity. For practical application, we propose several sequential algorithms to compute appropriate wavelength assignments. We also perform computational experiments to evaluate their performance. For the iterative algorithms, we identify characteristic patterns of progression. Two of these algorithms qualify for application in practice.
Chvatal-Gomory cuts are among the most well-known classes of cutting planes for general integer linear programs (ILPs). In case the constraint multipliers are either 0 or $\frac{1}{2}$, such cuts are known as $\{0,\frac{1}{2}\}$-cuts. It has been proven by Caprara and Fischetti (1996) that separation of $\{0,\frac{1}{2}\}$-cuts is NP-hard. In this paper, we study ways to separate $\{0,\frac{1}{2}\}$-cuts effectively in practice. We propose a range of preprocessing rules to reduce the size of the separation problem. The core of the preprocessing builds a Gaussian elimination-like procedure. To separate the most violated $\{0,\frac{1}{2}\}$-cut, we formulate the (reduced) problem as integer linear program. Some simple heuristic separation routines complete the algorithmic framework. Computational experiments on benchmark instances show that the combination of preprocessing with exact and/or heuristic separation is a very vital idea to generate strong generic cutting planes for integer linear programs and to reduce the overall computation times of state-of-the-art ILP-solvers.
In this paper, we investigate the connection availabilities for the new protection scheme Demand-wise Shared Protection (DSP) and describe an appropriate approach for their computation. The exemplary case study on two realistic network scenarios shows that in most cases the availabilities for DSP are comparable with that for 1+1 path protection and better than in case of shared path protection.