@misc{BodlaenderKosterEijkhofetal.2001, 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}, year = {2001}, 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{HoeselKosterLeenseletal.2000, author = {Hoesel, Stan P.M. van and Koster, Arie M.C.A. and Leensel, Robert L.M.J. van de and Savelsbergh, Martin W.P.}, title = {Polyhedral Results for the Edge Capacity Polytope}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5907}, number = {00-22}, year = {2000}, abstract = {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.}, language = {en} } @misc{BodlaenderFominKosteretal.2006, author = {Bodlaender, Hans L. and Fomin, Fedor V. and Koster, Arie M.C.A. and Kratsch, Dieter and Thilikos, Dimitrios M.}, title = {On exact algorithms for treewidth}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9265}, number = {06-32}, year = {2006}, abstract = {We give experimental and theoretical results on the problem of computing the treewidth of a graph by exact exponential time algorithms using exponential space or using only polynomial space. We first report on an implementation of a dynamic programming algorithm for computing the treewidth of a graph with running time \$O^\ast(2^n)\$. This algorithm is based on the old dynamic programming method introduced by Held and Karp for the {\sc Tra veling Salesman} problem. We use some optimizations that do not affect the worst case running time but improve on the running time on actual instances and can be seen to be practical for small instances. However, our experiments show that the space use d by the algorithm is an important factor to what input sizes the algorithm is effective. For this purpose, we settle the problem of computing treewidth under the restriction that the space used is only polynomial. In this direction we give a simple \$O^\ast(4^n)\$ al gorithm that requires {\em polynomial} space. We also show that with a more complicated algorithm, using balanced separators, {\sc Treewidth} can be computed in \$O^\ast(2.9512^n)\$ time and polynomial space.}, language = {en} } @misc{KosterBodlaenderHoesel2001, 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}, year = {2001}, 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{AardalHoeselKosteretal.2001, 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}, year = {2001}, 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{KosterZymolka2004, author = {Koster, Arie M.C.A. and Zymolka, Adrian}, title = {Provably Good Solutions for Wavelength Assignment in Optical Networks}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8155}, number = {04-40}, year = {2004}, abstract = {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.}, language = {en} } @misc{KosterZymolka2004, author = {Koster, Arie M.C.A. and Zymolka, Adrian}, title = {Linear Programming Lower Bounds for Minimum Converter Wavelength Assignment in Optical Networks}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8160}, number = {04-41}, year = {2004}, abstract = {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.}, language = {en} } @misc{WolleKosterBodlaender2004, author = {Wolle, Thomas and Koster, Arie M.C.A. and Bodlaender, Hans L.}, title = {A Note on Contraction Degeneracy}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8180}, number = {04-43}, year = {2004}, abstract = {The parameter contraction degeneracy -- the maximum minimum degree over all minors of a graph -- is a treewidth lower bound and was first defined in (Bodlaender, Koster, Wolle, 2004). In experiments it was shown that this lower bound improves upon other treewidth lower bounds. In this note, we examine some relationships between the contraction degeneracy and connected components of a graph, block s of a graph and the genus of a graph. We also look at chordal graphs, and we study an upper bound on the contraction degeneracy and another lower bound for treewidth. A data structure that can be used for algorithms computing the degeneracy and similar parameters, is also described.}, language = {en} } @misc{KosterWolleBodlaender2004, author = {Koster, Arie M.C.A. and Wolle, Thomas and Bodlaender, Hans L.}, title = {Degree-Based Treewidth Lower Bounds}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8193}, number = {04-44}, year = {2004}, abstract = {Every lower bound for treewidth can be extended by taking the maximum of the lower bound over all subgraphs or minors. This extension is shown to be a very vital idea for improving treewidth lower bounds. In this paper, we investigate a total of nine graph parameters, providing lower bounds for treewidth. The parameters have in common that they all are the vertex-degree of some vertex in a subgra ph or minor of the input graph. We show relations between these graph parameters and study their computational complexity. To allow a practical comparison of the bounds, we developed heuristic algorithms for those parameters that are NP-hard to compute. Computational experiments show that combining the treewidth lower bounds with minors can considerably improve the lower bounds.}, language = {en} } @misc{BodlaenderKoster2004, 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}, year = {2004}, 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{OrlowskiKosterRaacketal.2006, author = {Orlowski, Sebastian and Koster, Arie M.C.A. and Raack, Christian and Wess{\"a}ly, Roland}, title = {Two-layer Network Design by Branch-and-Cut featuring MIP-based Heuristics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9412}, number = {06-47}, year = {2006}, abstract = {This paper deals with MIP-based primal heuristics to be used within a branch-and-cut approach for solving multi-layer telecommunication network design problems. Based on a mixed-integer programming formulation for two network layers, we present three heuristics for solving important subproblems, two of which solve a sub-MIP. On multi-layer planning instances with many parallel logical links, we show the effectiveness of our heuristics in finding good solutions early in the branch-and-cut search tree.}, language = {en} } @misc{HuelsermannJaegerKosteretal.2006, author = {H{\"u}lsermann, Ralf and J{\"a}ger, Monika and Koster, Arie M.C.A. and Orlowski, Sebastian and Wess{\"a}ly, Roland and Zymolka, Adrian}, title = {Availability and Cost Based Evaluation of Demand-wise Shared Protection}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9080}, number = {06-15}, year = {2006}, abstract = {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.}, language = {en} } @misc{KosterWagler2005, author = {Koster, Arie M.C.A. and Wagler, Annegret}, title = {Comparing Imperfection Ratio and Imperfection Index for Graph Classes}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8836}, number = {05-50}, year = {2005}, abstract = {Perfect graphs constitute a well-studied graph class with a rich structure, reflected by many characterizations with respect to different concepts. Perfect graphs are, for instance, precisely those graphs \$G\$ where the stable set polytope \$STAB(G)\$ coincides with the fractional stable set polytope \$QSTAB(G)\$. For all imperfect graphs \$G\$ it holds that \$STAB(G) \subset QSTAB(G)\$. It is, therefore, natural to use the difference between the two polytopes in order to decide how far an imperfect graph is away from being perfect; we discuss three different concepts, involving the facet set of \$STAB( G)\$, the disjunctive index of \$QSTAB(G)\$, and the dilation ratio of the two polytopes. Including only certain types of facets for \$STAB(G)\$, we obtain graphs that are in some sense close to perfect graphs, for example minimally immperfect graphs, and certain other classes of so-called rank-perfect graphs. The imperfection ratio has been introduced by (Gerke and McDiarmid, 2001) as the dilation ratio of \$STAB(G)\$ and \$QSTAB(G)\$, whereas (Aguilera et al., 2003) suggest to take the disjunctive index of \$Q STAB(G)\$ as the imperfection index of \$G\$. For both invariants there exist no general upper bounds, but there are bounds known for the imperfection ratio of several graph classes (Coulonges et al. 2005, Gerke and McDiarmid, 2001). Outgoing from a graph-theoretical interpretation of the imperfection index, we conclude that the imperfection index is NP-hard to compute and we prove that there exists no upper bound on the imperfect ion index for those graph classes with a known bounded imperfection ratio. Comparing the two invariants on those classes, it seems that the imperfection index measures imperfection much more roughly than the imperfection ratio; therefoe, discuss possible directions for refinements.}, language = {en} } @misc{BodlaenderGrigorievKoster2005, author = {Bodlaender, Hans L. and Grigoriev, Alexander and Koster, Arie M.C.A.}, title = {Treewidth Lower Bounds with Brambles}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8878}, number = {05-54}, year = {2005}, abstract = {In this paper we present a new technique for computing lower bounds for graph treewidth. Our technique is based on the fact that the treewidth of a graph \$G\$ is the maximum order of a bramble of \$G\$ minus one. We give two algorithms: one for general graphs, and one for planar graphs. The algorithm for planar graphs is shown to give a lower bound for both the treewidth and branchwidth that is at most a constant factor away from the optimum. For both algorithms, we report on extensive computational experiments that show that the algorithms give often excellent lower bounds, in particular when applied to (close to) planar graphs.}, language = {en} } @misc{GruberKosterOrlowskietal.2005, author = {Gruber, Claus G. and Koster, Arie M.C.A. and Orlowski, Sebastian and Wess{\"a}ly, Roland and Zymolka, Adrian}, title = {A new model and a computational study for Demand-wise Shared Protection}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8880}, number = {05-55}, year = {2005}, abstract = {This report combines the contributions to INOC 2005 (Wess{\"a}lly et al., 2005) and DRCN 2005 (Gruber et al., 2005). A new integer linear programming model for the end-to-end survivability concept deman d-wise shared protection (DSP) is presented. DSP is based on the idea that backup capacity is dedicated to a particular demand, but shared within a demand. It combines advantages of dedicated and shared protection: It is more cost-efficient than dedicated protection and operationally easier than shared protection. In a previous model for DSP, the number of working and backup paths to be configured for a particular demand has been an input parameter; in the more general model for DSP investigated in this paper, this value is part of the decisions to take. To use the new DSP model algorithmically, we suggest a branch-and-cut approach which employs a column generation procedure to deal with the exponential number of routing variables. A computational study to compare the new resilience mechanism DSP with dedicated and shared path protection is performed. The results for five realistic network planning scenarios reveal that the best solutions for DSP are on average 15\\% percent better than the corresponding 1+1 dedicated path protection solutions, and only 15\\% percent worse than shared path protection.}, language = {en} } @misc{KosterZymolka2000, author = {Koster, Arie M.C.A. and Zymolka, Adrian}, title = {Stable Multi-Sets}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6047}, number = {00-36}, year = {2000}, abstract = {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.}, language = {en} } @misc{BleyKosterKroelleretal.2003, 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}, year = {2003}, 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{BodlaenderKoster2003, 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}, year = {2003}, 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{KosterZymolka2003, 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}, year = {2003}, 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{EijkhofBodlaenderKoster2002, 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}, year = {2002}, 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} }