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#### Institute

In this paper we describe the results of a computational study towards the (re)optimization of signaling transfer points (STPs) in telecommunication networks. The best performance of an STP is achieved whenever the traffic load is evenly distributed among the internal components. Due to the continuously changing traffic pattern, the load of the components has to be re-optimized on a regular basis. Besides the balancing objective also the number of rearrangements have to be taken into account. In this paper we present two alternative formulations to deal with both requirements. Computational results show that for both formulations (near) optimal solutions can be obtained within reasonable time limits.

Network loading problems occur in the design of telecommunication networks, in many different settings. The polyhedral structure of this problem is important in developing solution methods for the problem. In this paper we investigate the polytope of the problem restricted to one edge of the network (the edge capacity problem). We describe classes of strong valid inequalities for the edge capacity polytope, and we derive conditions under which these constraints define facets. As the edge capacity problem is a relaxation of the network loading problem, their polytopes are intimately related. We, therefore, also give conditions under which the inequalities of the edge capacity polytope define facets of the network loading polytope. Furthermore, some structural properties are derived, such as the relation of the edge capacity polytope to the knapsack polytope. We conclude the theoretical part of this paper with some lifting theorems, where we show that this problem is polynomially solvable for most of our classes of valid inequalities. In a computational study the quality of the constraints is investigated. Here, we show that the valid inequalities of the edge capacity polytope are not only important for solving the edge capacity problem, but also for the network loading problem, showing that the edge capacity problem is an important subproblem.

Stable Multi-Sets
(2000)

In this paper we introduce a generalization of stable sets: stable multi-sets. A stable multi-set is an assignment of integers to the vertices of a graph, such that specified bounds on vertices and edges are not exceeded. In case all vertex and edge bounds equal one, stable multi-sets are equivalent to stable sets. For the stable multi-set problem, we derive reduction rules and study the associated polytope. We state necessary and sufficient conditions for the extreme points of the linear relaxation to be integer. These conditions generalize the conditions for the stable set polytope. Moreover, the classes of odd cycle and clique inequalities for stable sets are generalized to stable multi-sets and conditions for them to be facet defining are determined. The study of stable multi-sets is initiated by optimization problems in the field of telecommunication networks. Stable multi-sets emerge as an important substructure in the design of optical networks.

Traffic in communication networks fluctuates heavily over time.
Thus, to avoid capacity bottlenecks, operators highly overestimate
the traffic volume during network planning. In this paper we
consider telecommunication network design under traffic uncertainty,
adapting the robust optimization approach of Bertsimas and Sim [2004]. We
present three different mathematical formulations for this problem,
provide valid inequalities, study the computational implications,
and evaluate the realized robustness.
To enhance the performance of the mixed-integer programming solver
we derive robust cutset inequalities generalizing their
deterministic counterparts. Instead of a single cutset inequality
for every network cut, we derive multiple valid
inequalities by exploiting the extra variables available in the
robust formulations. We show that these inequalities define facets
under certain conditions and that they completely describe a projection
of the robust cutset polyhedron if the cutset consists of a single edge.
For realistic networks and live traffic measurements we compare the
formulations and report on the speed up by the valid inequalities.
We study the "price of robustness" and evaluate the
approach by analyzing the real network load. The results show that
the robust optimization approach has the potential to support
network planners better than present methods.

Frequenzplanung im Mobilfunk
(2002)

Telekommunikation ist seit Jahren \glqq in\grqq. Zunächst gab es einen enormen Aufschwung; neue Technologien und Dienste haben eine überwältigende, nicht vorhersehbare Akzeptanz gefunden. Derzeit ist -- ausgelöst durch die UMTS-Lizenzversteigerungen, Rezessions- und Sättigungstendenzen -- eine Krise zu verzeichnen. Viele (auch wir) sind davon überzeugt, dass technischer Fortschritt und nützliche Dienste demnächst die Stimmung wieder ändern werden. Wenigen ist allerdings bewusst, welche Rolle Mathematik bei der Entwicklung und dem effizienten Einsatz vieler der neuen Kommunikationstechnologien spielt. In diesem Artikel soll kein Überblick über diesen umfangreichen Themenkreis gegeben werden. Wir zeigen lediglich an einem konkreten Beispiel aus dem Mobilfunk, der Frequenzplanung in GSM-Funknetzen, was man durch geeignete Modellierung der praktischen Fragestellung und den Einsatz problemadäquater Algorithmen erreichen kann.

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.

Several sets of reductions rules are known for preprocessing a graph when computing its treewidth. In this paper, we give reduction rules for a weighted variant of treewidth, motivated by the analysis of algorithms for probabilistic networks. We present two general reduction rules that are safe for weighted treewidth. They generalise many of the existing reduction rules for treewidth. Experimental results show that these reduction rules can significantly reduce the problem size for several instances of real-life probabilistic networks.

Signaling is crucial to the operation of modern telecommunication networks. A breakdown in the signaling infrastructure typically causes customer service failures, incurs revenue losses, and hampers the company image. Therefore, the signaling network has to be highest reliability and survivability. This in particular holds for the routers in such a network, called \textit{signaling transfer points\/} (STPs). The robustness of an STP can be improved by equally distributing the load over the internal processing units. Several constraints have to be taken into account. The load of the links connected to a processing unit changes over time introducing an imbalance of the load. In this paper, we show how integer linear programming can be applied to reduce the imbalance within an STP, while keeping the number of changes small. Two alternative models are presented. Computational experiments validate the integer programming approach in practice. The GSM network operator E-Plus saves substantial amounts of time and money by employing the proposed approach.

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.