@misc{BodlaenderKoster, author = {Bodlaender, Hans L. and Koster, Arie M.C.A.}, title = {On the Maximum Cardinality Search Lower Bound for Treewidth}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8201}, number = {04-45}, abstract = {The Maximum Cardinality Search algorithm visits the vertices of a graph in some order, such that at each step, an unvisited vertex that has the largest number of visited neighbors becomes visited. An MCS-ordering of a graph is an ordering of the vertices that can be generated by the Maximum Cardinality Search algorithm. The visited degree of a vertex \$v\$ in an MCS-ordering is the number of neighbors of \$v\$ that are before \$v\$ in the ordering. The visited degree of an MCS-ordering \$\psi\$ of \$G\$ is the maximum visited degree over all vertices \$v\$ in \$\psi\$. The maximum visited degree over all MCS-orderings of graph \$G\$ is called its {\em maximum visited degree}. Lucena (2003) showed that the treewidth of a graph \$G\$ is at least its maximum visited degree. We show that the maximum visited degree is of size \$O(\log n)\$ for planar graphs, and give examples of planar graphs \$G\$ with maximum visited degree \$k\$ with \$O(k!)\$ vertices, for all \$k\in \Bbb{N}\$. Given a graph \$G\$, it is NP-complete to determine if its maximum visited degree is at least \$k\$, for any fixed \$k\geq 7\$. Also, this problem does not have a polynomial time approximation algorithm with constant ratio, unless P=NP. Variants of the problem are also shown to be NP-complete. We also propose and experimentally analyses some heuristics for the problem. Several tiebreakers for the MCS algorithm are proposed and evaluated. We also give heuristics that give upper bounds on the value of the maximum visited degree of a graph, which appear to give results close to optimal on many graphs from real life applications.}, language = {en} } @misc{BleyKosterKroelleretal., author = {Bley, Andreas and Koster, Arie M.C.A. and Kr{\"o}ller, Alexander and Wess{\"a}ly, Roland and Zymolka, Adrian}, title = {Kosten- und Qualit{\"a}tsoptimierung in Kommunikationsnetzen}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7537}, number = {03-31}, abstract = {Der scharfe Wettbewerb innerhalb der Telekommunikationsbranche zwingt die Netzbetreiber dazu, ihre Investitionen genau zu planen und immer wieder Einsparungsmanahmen durchzuf{\"u}hren. Gleichzeitig ist es jedoch wichtig, die Qualit{\"a}t der angebotenen Dienste zu verbessern, um neue Kunden zu gewinnen und langfristig an sich zu binden. Die mathematische Optimierung bietet sich f{\"u}r viele solcher Aufgabenstellungen als hervorragend geeignetes Planungswerkzeug an. Ziel dieses Artikels ist es, ihre Methodik und ihre Anwendung speziell zur Kosten- und Qualit{\"a}tsoptimierung in Kommunikationsnetzen vorzustellen. Anhand von vier konkreten Planungsaufgaben aus dem Bereich der Festnetzplanung wird aufgezeigt, wie sich komplexe Zusammenh{\"a}nge in flexiblen mathematischen Modellen abbilden lassen und welche Verfahren zur automatisierten Bearbeitung der Probleme eingesetzt werden k{\"o}nnen. Die hier vorgestellten Methoden zeichnen sich insbesondere dadurch aus, dass sie neben hochwertigen L{\"o}sungen auch eine Qualittsgarantie liefern, mit der sich die Lsungen fundiert bewerten lassen. Die dokumentierten Ergebnisse aus verschiedenen Industrieprojekten belegen die Eignung und G{\"u}te der mathematischen Optimierung f{\"u}r die Praxis.}, language = {de} } @misc{BodlaenderKoster, author = {Bodlaender, Hans L. and Koster, Arie M.C.A.}, title = {Safe separators for treewidth}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7544}, number = {03-32}, abstract = {A set of vertices \$S\subseteq V\$ is called a safe separator for treewidth, if \$S\$ is a separator of \$G\$, and the treewidth of \$G\$ equals the maximum of the treewidth over all connected components \$W\$ of \$G-S\$ of the graph, obtained by making \$S\$ a clique in the subgraph of \$G\$, induced by \$W\cup S\$. We show that such safe separators are a very powerful tool for preprocessing graphs when we want to compute their treewidth. We give several sufficient conditions for separators to be safe, allowing such separators, if existing, to be found in polynomial time. In particular, every minimal separator of size one or two is safe, every minimal separator of size three that does not split off a component with only one vertex is safe, and every minimal separator that is an almost clique is safe; an almost clique is a set of vertices \$W\$ such that there is a \$v\in W\$ with \$W-\{v\}\$ a clique. We report on experiments that show significant reductions of instance sizes for graphs from proba! bilistic networks and frequency assignment.}, language = {en} } @misc{KosterBodlaenderHoesel, author = {Koster, Arie M.C.A. and Bodlaender, Hans L. and Hoesel, Stan P.M. van}, title = {Treewidth: Computational Experiments}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6644}, number = {01-38}, abstract = {Many {\cal NP}-hard graph problems can be solved in polynomial time for graphs with bounded treewidth. Equivalent results are known for pathwidth and branchwidth. In recent years, several studies have shown that this result is not only of theoretical interest but can successfully be applied to find (almost) optimal solutions or lower bounds for diverse optimization problems. To apply a tree decomposition approach, the treewidth of the graph has to be determined, independently of the application at hand. Although for fixed \$k\$, linear time algorithms exist to solve the decision problem ``treewidth \$\leq k\$'', their practical use is very limited. The computational tractability of treewidth has been rarely studied so far. In this paper, we compare four heuristics and two lower bounds for instances from applications such as the frequency assignment problem and the vertex coloring problem. Three of the heuristics are based on well-known algorithms to recognize triangulated graphs. The fourth heuristic recursively improves a tree decomposition by the computation of minimal separating vertex sets in subgraphs. Lower bounds can be computed from maximal cliques and the minimum degree of induced subgraphs. A computational analysis shows that the treewidth of several graphs can be identified by these methods. For other graphs, however, more sophisticated techniques are necessary.}, language = {en} } @misc{BodlaenderKosterEijkhofetal., author = {Bodlaender, Hans L. and Koster, Arie M.C.A. and Eijkhof, Frank van den and Gaag, Linda C. van der}, title = {Pre-processing for Triangulation of Probabilistic Networks}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6655}, number = {01-39}, abstract = {The currently most efficient algorithm for inference with a probabilistic network builds upon a triangulation of a network's graph. In this paper, we show that pre-processing can help in finding good triangulations for probabilistic networks, that is, triangulations with a minimal maximum clique size. We provide a set of rules for stepwise reducing a graph, without losing optimality. This reduction allows us to solve the triangulation problem on a smaller graph. From the smaller graph's triangulation, a triangulation of the original graph is obtained by reversing the reduction steps. Our experimental results show that the graphs of some well-known real-life probabilistic networks can be triangulated optimally just by preprocessing; for other networks, huge reductions in their graph's size are obtained.}, language = {en} } @misc{AardalHoeselKosteretal., author = {Aardal, Karen I. and Hoesel, Stan P.M. van and Koster, Arie M.C.A. and Mannino, Carlo and Sassano, Antonio}, title = {Models and Solution Techniques for Frequency Assignment Problems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6667}, number = {01-40}, abstract = {{\begin{rawhtml} Revised Version unter http://dx.doi.org/10.1007/s10479-007-0178-0 \end{rawhtml}} Wireless communication is used in many different situations such as mobile telephony, radio and TV broadcasting, satellite communication, and military operations. In each of these situations a frequency assignment problem arises with application specific characteristics. Researchers have developed different modelling ideas for each of the features of the problem, such as the handling of interference among radio signals, the availability of frequencies, and the optimization criterion. This survey gives an overview of the models and methods that the literature provides on the topic. We present a broad description of the practical settings in which frequency assignment is applied. We also present a classification of the different models and formulations described in the literature, such that the common features of the models are emphasized. The solution methods are divided in two parts. Optimization and lower bounding techniques on the one hand, and heuristic search techniques on the other hand. The literature is classified according to the used methods. Again, we emphasize the common features, used in the different papers. The quality of the solution methods is compared, whenever possible, on publicly available benchmark instances.}, language = {en} } @misc{EijkhofBodlaenderKoster, author = {Eijkhof, Frank van den and Bodlaender, Hans L. and Koster, Arie M.C.A.}, title = {Safe reduction rules for weighted treewidth}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7164}, number = {02-49}, abstract = {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.}, language = {en} } @misc{EisenblaetterKosterWallbaumetal., author = {Eisenbl{\"a}tter, Andreas and Koster, Arie M.C.A. and Wallbaum, Randolf and Wess{\"a}ly, Roland}, title = {Load Balancing in Signaling Transfer Points}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7179}, number = {02-50}, abstract = {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.}, language = {en} } @misc{KosterZymolka, author = {Koster, Arie M.C.A. and Zymolka, Adrian}, title = {Polyhedral Investigations on Stable Multi-Sets}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7324}, number = {03-10}, abstract = {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.}, language = {en} } @misc{KosterZymolkaJaegeretal., author = {Koster, Arie M.C.A. and Zymolka, Adrian and J{\"a}ger, Monika and H{\"u}lsermann, Ralf and Gerlach, Christoph}, title = {Demand-wise Shared Protection for Meshed Optical Networks}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7461}, number = {03-24}, abstract = {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.}, language = {en} }