TY - GEN A1 - Pfetsch, Marc A1 - Borndörfer, Ralf T1 - Routing in Line Planning for Public Transportation N2 - The line planning problem is one of the fundamental problems in strategic planning of public and rail transport. It consists in finding lines and corresponding frequencies in a network such that a giv en demand can be satisfied. There are two objectives. Passengers want to minimize travel times, the transport company wishes to minimize operating costs. We investigate three variants of a multi-commo dity flow model for line planning that differ with respect to passenger routings. The first model allows arbitrary routings, the second only unsplittable routings, and the third only shortest path rou tings with respect to the network. We compare these models theoretically and computationally on data for the city of Potsdam. T3 - ZIB-Report - 05-36 KW - line planning KW - column generation KW - passenger routing Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8706 ER - TY - GEN A1 - Bosse, Hartwig A1 - Grötschel, Martin A1 - Henk, Martin T1 - Polynomial Inequalities Representing Polyhedra N2 - Our main result is that every $n$-dimensional polytope can be described by at most $2n-1$ polynomial inequalities and, moreover, these polynomials can explicitly be constructed. For an $n$-dimensional pointed polyhedral cone we prove the bound $2n-2$ and for arbitrary polyhedra we get a constructible representation by $2n$ polynomial inequalities. T3 - ZIB-Report - 04-53 KW - polyhedra and polytopes KW - semi-algebraic sets KW - polyhedral combinatorics KW - polynomial inequalities KW - stability index Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8284 ER - TY - GEN A1 - Stephan, Rüdiger T1 - Facets of the (s,t)-p-path polytope N2 - \noindent We give a partial description of the $(s,t)-p$-path polytope of a directed graph $D$ which is the convex hull of the incidence vectors of simple directed $(s,t)$-paths in $D$ of length $p$. First, we point out how the $(s,t)-p$-path polytope is located in the family of path and cycle polyhedra. Next, we give some classes of valid inequalities which are very similar to inequalities which are valid for the $p$-cycle polytope, that is, the convex hull of the incidence vectors of simple cycles of length $p$ in $D$. We give necessary and sufficient conditions for these inequalities to be facet defining. Furthermore, we consider a class of inequalities that has been identifie d to be valid for $(s,t)$-paths of cardinality at most $p$. Finally, we transfer the results to related polytopes, in particular, the undirected counterpart of the $(s,t)-p$-path polytope. T3 - ZIB-Report - 06-38 Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9328 ER - TY - GEN A1 - Joswig, Michael A1 - Pfetsch, Marc T1 - Computing Optimal Morse Matchings N2 - Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NP-hard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results. T3 - ZIB-Report - 04-37 KW - discrete Morse function KW - Morse matching Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8120 ER - TY - GEN A1 - Jokar, Sadegh A1 - Pfetsch, Marc T1 - Exact and Approximate Sparse Solutions of Underdetermined Linear Equations N2 - In this paper, we empirically investigate the NP-hard problem of finding sparse solutions to linear equation systems, i.e., solutions with as few nonzeros as possible. This problem has received considerable interest in the sparse approximation and signal processing literature, recently. We use a branch-and-cut approach via the maximum feasible subsystem problem to compute optimal solutions for small instances and investigate the uniqueness of the optimal solutions. We furthermore discuss five (modifications of) heuristics for this problem that appear in different parts of the literature. For small instances, the exact optimal solutions allow us to evaluate the quality of the heuristics, while for larger instances we compare their relative performance. One outcome is that the basis pursuit heuristic performs worse, compared to the other methods. Among the best heuristics are a method due to Mangasarian and a bilinear approach. T3 - ZIB-Report - 07-05 KW - sparse representations KW - basis pursuit KW - orthogonal matching pursuit KW - maximum feasible subsystem problem Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9488 ER - TY - GEN A1 - Bley, Andreas T1 - Approximability of Unsplittable Shortest Path Routing Problems N2 - In this paper, we discuss the relation of unsplittable shortest path routing (USPR) to other routing schemes and study the approximability of three USPR network planning problems. Given a digraph $D=(V,A)$ and a set $K$ of directed commodities, an USPR is a set of flow paths $\Phi_{(s,t)}$, $(s,t)\in K$, such that there exists a metric $\lambda=(\lambda_a)\in \mathbb{Z}^A_+$ with respect to which each $\Phi_{(s,t)}$ is the unique shortest $(s,t)$-path. In the \textsc{Min-Con-USPR} problem, we seek for an USPR that minimizes the maximum congestion over all arcs. We show that this problem is hard to approximate within a factor of $\mathcal{O}(|V|^{1-\epsilon})$, but easily approximable within min$(|A|,|K|)$ in general and within $\mathcal{O}(1)$ if the underlying graph is an undirected cycle or a bidirected ring. We also construct examples where the minimum congestion that can be obtained by USPR is a factor of $\Omega(|V|^2)$ larger than that achievable by unsplittable flow routing or by shortest multi-path routing, and a factor of $\Omega(|V|)$ larger than by unsplittable source-invariant routing. In the CAP-USPR problem, we seek for a minimum cost installation of integer arc capacities that admit an USPR of the given commodities. We prove that this problem is $\mathcal{NP}$-hard to approximate within $2-\epsilon$ (even in the undirected case), and we devise approximation algorithms for various special cases. The fixed charge network design problem \textsc{Cap-USPR}, where the task is to find a minimum cost subgraph of $D$ whose fixed arc capacities admit an USPR of the commodities, is shown to be $\mathcal{NPO}$-complete. All three problems are of great practical interest in the planning of telecommunication networks that are based on shortest path routing protocols. Our results indicate that they are harder than the corresponding unsplittable flow or shortest multi-path routing problems. T3 - ZIB-Report - 06-02 KW - Shortest path routing KW - unsplittable flow KW - computational complexity KW - approximation Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8968 ER - TY - GEN A1 - Shioura, Akiyoshi T1 - Note on L#-convex Function Minimization Algorithms: Comparison of Murota's and Kolmogorov's Algorithms N2 - The concept of L##-convexity is introduced by Fujishige--Murota (2000) as a discrete convexity for functions defined over the integer lattice. The main aim of this note is to understand the difference of the two algorithms for L##-convex function minimization: Murota's steepest descent algorithm (2003) and Kolmogorov's primal algorithm (2005). T3 - ZIB-Report - 06-03 Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8979 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Grötschel, Martin A1 - Pfetsch, Marc T1 - A Column-Generation Approach to Line Planning in Public Transport N2 - The line planning problem is one of the fundamental problems in strategic planning of public and rail transport. It consists in finding lines and corresponding frequencies in a transport network such that a given travel demand can be satisfied. There are (at least) two objectives. The transport company wishes to minimize operating costs, the passengers want to minimize travel times. We propose a n ew multi-commodity flow model for line planning. Its main features, in comparison to existing models, are that the passenger paths can be freely routed and that the lines are generated dynamically. We discuss properties of this model and investigate its complexity. Results with data for the city of Potsdam, Germany, are reported. T3 - ZIB-Report - 05-18 KW - line planning KW - column generation Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8522 ER - TY - GEN A1 - Grötschel, Martin T1 - Schnelle Rundreisen: Das Travelling Salesman-Problem N2 - Das Travelling-Salesman-Problem (TSP) ist das am intensivsten untersuchte kombinatorische Optimierungsproblem. In diesem Abschnitt wird eine Einführung in das TSP gegeben. Es werden Problemstellungen erläutert, Anwendungen skizziert und einige Schwierigkeiten bei der korrekten Modellierung der Zielfunktion dargelegt. Es ist gar nicht so klar, was in einem konkreten Problem die wirkliche Entfernung ist. Exakte und approximative Lösungsverfahren werden an Beispielen skizziert, und es wird angedeutet, dass man, obwohl TSPs zu den theoretisch schweren Problemen zählen, in der Praxis TSPs von atemberaubender Größe lösen kann. T3 - ZIB-Report - 05-57 KW - Travelling-Salesman-Problem KW - TSP KW - Rundreisen KW - optimale Tour Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8901 ER - TY - GEN A1 - Shioura, Akiyoshi A1 - Tanaka, Ken'ichiro T1 - Polynomial-Time Algorithms for Linear and Convex Optimization on Jump Systems N2 - The concept of jump system, introduced by Buchet and Cunningham (1995), is a set of integer points with a certain exchange property. In this paper, we discuss several linear and convex optimization problems on jump systems and show that these problems can be solved in polynomial time under the assumption that a membership oracle for a jump system is available. We firstly present a polynomial-time implementation of the greedy algorithm for the minimization of a linear function. We then consider the minimization of a separable-convex function on a jump system, and propose the first polynomial-time algorithm for this problem. The algorithm is based on the domain reduction approach developed in Shioura (1998). We finally consider the concept of M-convex functions on constant-parity jump systems which has been recently proposed by Murota (2006). It is shown that the minimization of an M-convex function can be solved in polynomial time by the domain reduction approach. T3 - ZIB-Report - 06-19 KW - jump system KW - discrete convex function KW - bisubmodular function KW - bisubmodular polyhedron KW - polynomial-time algorithm Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9122 ER - TY - GEN A1 - Kaibel, Volker A1 - Peinhardt, Matthias T1 - On the Bottleneck Shortest Path Problem N2 - The Bottleneck Shortest Path Problem is a basic problem in network optimization. The goal is to determine the limiting capacity of any path between two specified vertices of the network. This is equivalent to determining the unsplittable maximum flow between the two vertices. In this note we analyze the complexity of the problem, its relation to the Shortest Path Problem, and the impact of the underlying machine/computation model. T3 - ZIB-Report - 06-22 KW - unsplittable flow KW - linear time algorithm KW - shortest path Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9160 ER - TY - GEN A1 - Bley, Andreas T1 - On the Complexity of Vertex-Disjoint Length-Restricted Path Problems N2 - Let $G=(V,E)$ be a simple graph and $s$ and $t$ be two distinct vertices of $G$. A path in $G$ is called $\ell$-bounded for some $\ell\in\mathbb{N}$, if it does not contain more than $\ell$ edges. We study the computational complexity of approximating the optimum value for two optimization problems of finding sets of vertex-disjoint $\ell$-bounded $s,t$-paths in $G$. First, we show that computing the maximum number of vertex-disjoint $\ell$-bounded $s,t$-paths is $\mathcal{AP\kern-1pt X}$--complete for any fixed length bound $\ell\geq 5$. Second, for a given number $k\in\mathbb{N}$, $1\leq k \leq |V|-1$, and non-negative weights on the edges of $G$, the problem of finding $k$ vertex-disjoint $\ell$-bounded $s,t$-paths with minimal total weight is proven to be $\mathcal{NPO}$--complete for any length bound $\ell\geq 5$. Furthermore, we show that, even if $G$ is complete, it is $\mathcal{NP}$--complete to approximate the optimal solution value of this problem within a factor of $2^{\langle\phi\rangle^\epsilon}$ for any constant $0<\epsilon<1$, where $\langle\phi\rangle$ denotes the encoding size of the given problem instance $\phi$. We prove that these results are tight in the sense that for lengths $\ell\leq 4$ both problems are polynomially solvable, assuming that the weights satisfy a generalized triangle inequality in the weighted problem. All results presented also hold for directed and non-simple graphs. For the analogous problems where the path length restriction is replaced by the condition that all paths must have length equal to $\ell$ or where vertex-disjointness is replaced by edge-disjointness we obtain similar results. T3 - ZIB-Report - SC-98-20 KW - disjoint paths KW - length bounded paths KW - approximation KW - reducibility KW - completeness Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3639 ER - TY - GEN A1 - Ascheuer, Norbert A1 - Grötschel, Martin A1 - Krumke, Sven A1 - Rambau, Jörg T1 - Combinatorial Online Optimization N2 - In ``classical'' optimization, all data of a problem instance are considered given. The standard theory and the usual algorithmic techniques apply to such cases only. Online optimization is different. Many decisions have to be made before all data are available. In addition, decisions once made cannot be changed. How should one act ``best'' in such an environment? In this paper we survey online problems coming up in combinatorial optimization. We first outline theoretical concepts, such as competitiveness against various adversaries, to analyze online problems and algorithms. The focus, however, lies on real-world applications. We report, in particular, on theoretical investigations and our practical experience with problems arising in transportation and the automatic handling of material. T3 - ZIB-Report - SC-98-24 KW - Online Optimization KW - competitiveness KW - combinatorial optimization KW - real-world problems Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3674 ER - TY - GEN A1 - Bley, Andreas A1 - Grötschel, Martin A1 - Wessäly, Roland T1 - Design of Broadband Virtual Private Networks: Model and Heuristics for the B–WiN N2 - We investigate the problem of designing survivable broadband virtual private networks that employ the Open Shortest Path First (OSPF) routing protocol to route the packages. The capacities available for the links of the network are a minimal capacity plus multiples of a unit capacity. Given the directed communication demands between all pairs of nodes, we wish to select the capacities in a such way, that even in case of a single node or a single link failure a specified percentage of each demand can be satisfied and the costs for these capacities are minimal. We present a mixed--integer linear programming formulation of this problem and several heuristics for its solution. Furthermore, we report on computational results with real-world data. T3 - ZIB-Report - SC-98-13 KW - Telecommunication Network Design KW - Survivable Networks KW - Network Capacity Planning KW - OSPF Routing KW - Shortest Path Routing KW - Heuristics Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3565 ER - TY - GEN A1 - Ascheuer, Norbert A1 - Krumke, Sven A1 - Rambau, Jörg T1 - The Online Transportation Problem: Competitive Scheduling of Elevators N2 - In this paper we consider the following online transportation problem (\textsc{Oltp}): Objects are to be transported between the vertices of a given graph. Transportation requests arrive online, specifying the objects to be transported and the corresponding source and target vertex. These requests are to be handled by a server which commences its work at a designated origin vertex and which picks up and drops objects at their starts and destinations. After the end of its service the server returns to its start. The goal of \textsc{Oltp} is to come up with a transportation schedule for the server which finishes as early as possible. We first show a lower bound of~$5/3$ for the competitive ratio of any deterministic algorithm. We then analyze two simple and natural strategies which we call \textsf{REPLAN} and \textsf{IGNORE}. \textsf{REPLAN} completely discards its schedule and recomputes a new one when a new request arrives. \textsf{IGNORE} always runs a (locally optimal) schedule for a set of known requests and ignores all new requests until this schedule is completed. We show that both strategies, \textsf{REPLAN} and \textsf{IGNORE}, are $5/2$-competitive. We also present a somewhat less natural strategy \textsf{SLEEP}, which in contrast to the other two strategies may leave the server idle from time to time although unserved requests are known. We also establish a competitive ratio of~$5/2$ for the algorithm \textsf{SLEEP}. Our results are extended to the case of ``open schedules'' where the server is not required to return to its start position at the end of its service. T3 - ZIB-Report - SC-98-34 KW - online optimization KW - competitive analysis KW - elevator Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3779 ER - TY - GEN A1 - Grötschel, Martin A1 - Hiller, Benjamin A1 - Tuchscherer, Andreas T1 - Combinatorial Online Optimization: Elevators & Yellow Angels N2 - In \emph{classical optimization} it is assumed that full information about the problem to be solved is given. This, in particular, includes that all data are at hand. The real world may not be so nice'' to optimizers. Some problem constraints may not be known, the data may be corrupted, or some data may not be available at the moments when decisions have to be made. The last issue is the subject of \emph{online optimization} which will be addressed here. We explain some theory that has been developed to cope with such situations and provide examples from practice where unavailable information is not the result of bad data handling but an inevitable phenomenon. T3 - ZIB-Report - 07-36 KW - Online-Optimierung KW - Aufzugssteuerung KW - Dispatching von Fahrzeugen KW - online optimization KW - elevator control KW - vehicle dispatching Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10360 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Karbstein, Marika T1 - A Primal-Dual Approximation Algorithm for the Steiner Connectivity Problem N2 - We extend the primal-dual approximation technique of Goemans and Williamson to the Steiner connectivity problem, a kind of Steiner tree problem in hypergraphs. This yields a (k+1)-approximation algorithm for the case that k is the minimum of the maximal number of nodes in a hyperedge minus 1 and the maximal number of terminal nodes in a hyperedge. These results require the proof of a degree property for terminal nodes in hypergraphs which generalizes the well-known graph property that the average degree of terminal nodes in Steiner trees is at most 2. T3 - ZIB-Report - 13-54 KW - Primal-Dual Approximation KW - Steiner Connectivity Problem KW - Degree Property KW - Hypergraph Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-42430 SN - 1438-0064 ER - TY - GEN A1 - Koster, Arie M.C.A. A1 - Zymolka, Adrian T1 - Stable Multi-Sets N2 - 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. T3 - ZIB-Report - 00-36 KW - stable multi-sets KW - polyhedral combinatorics Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6047 ER - TY - GEN A1 - Koch, Thorsten A1 - Martin, Alexander A1 - Voß, Stefan T1 - SteinLib: An Updated Library on Steiner Tree Problems in Graphs N2 - In this paper we present the {\em SteinLib}, a library of data sets for the Steiner tree problem in graphs. This library extends former libraries on Steiner tree problems by many new interesting and difficult instances, most of them arising from real-world applications. We give a survey on the difficulty of these problem instances by giving references to state-of-the-art software packages that were the first or are currently among the best to solve these instances. T3 - ZIB-Report - 00-37 KW - Steiner Trees KW - Mathematical Programming Testdata Library Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6056 ER - TY - GEN A1 - Bley, Andreas T1 - A Lagrangian Approach for Integrated Network Design and Routing in IP Networks N2 - We consider the problem of designing a network that employs a non-bifurcated shortest path routing protocol. The network's nodes and the set of potential links are given together with a set of forecasted end-to-end traffic demands. All relevant hardware components installable at links or nodes are considered. The goal is to simultaneously choose the network's topology, to decide which hardware components to install on which links and nodes, and to find appropriate routing weights such that the overall network cost is minimized. In this paper, we present a mathematical optimization model for this problem and an algorithmic solution approach based on a Lagrangian relaxation. Computational results achieved with this approach for several real-world network planning problems are reported. T3 - ZIB-Report - 03-29 KW - Network Planning KW - Routing KW - IP Routing KW - Lagrangian relaxation Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7515 ER - TY - GEN A1 - Koster, Arie M.C.A. A1 - Zymolka, Adrian T1 - Polyhedral Investigations on Stable Multi-Sets N2 - 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. T3 - ZIB-Report - 03-10 KW - stable multi-sets KW - polyhedral combinatorics Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7324 ER - TY - GEN A1 - Hülsermann, Ralf A1 - Jäger, Monika A1 - Krumke, Sven A1 - Poensgen, Diana A1 - Rambau, Jörg A1 - Tuchscherer, Andreas T1 - Dynamic Routing Algorithms in Transparent Optical Networks An Experimental Study Based on Real Data N2 - Today's telecommunication networks are configured statically. Whenever a connection is established, the customer has permanent access to it. However, it is observed that usually the connection is not used continuously. At this point, dynamic provisioning could increase the utilization of network resources. WDM based Optical Transport Networks (OTNs) will shortly allow for fast dynamic network reconfiguration. This enables optical broadband leased line services on demand. Since service requests competing for network resources may lead to service blocking, it is vital to use appropriate strategies for routing and wavelength assignment in transparent optical networks. We simulate the service blocking probabilities of various dynamic algorithms for this problem using a well-founded traffic model for two realistic networks. One of the algorithms using shortest path routings performs best on all instances. Surprisingly, the tie-breaking rule between equally short paths in different wavelengths decides between success or failure. T3 - ZIB-Report - 02-35 KW - Dynamic Network Configuration KW - Routing and Wavelength Allocation KW - Transparent Optical Networks KW - Blocking Probability KW - Simulation Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7025 ER - TY - GEN A1 - Krumke, Sven A1 - Laura, Luigi A1 - Lipmann, Maarten A1 - Marchetti-Spaccamela, Alberto A1 - Paepe, Willem de A1 - Poensgen, Diana A1 - Stougie, Leen T1 - Non-Abusiveness Helps: An O(1)-Competitive Algorithm for Minimizing the Maximum Flow Time in the Online Traveling Salesman Problem N2 - In the online traveling salesman problem $OLTSP$ requests for visits to cities arrive online while the salesman is traveling. We study the $F{\_max}-OLTSP$ where the objective is to minimize the maximum flow time. This objective is particularly interesting for applications. Unfortunately, there can be no competitive algorithm, neither deterministic nor randomized. Hence, competitive analysis fails to distinguish online algorithms. Not even resource augmentation which is helpful in scheduling works as a remedy. This unsatisfactory situation motivates the search for alternative analysis methods. We introduce a natural restriction on the adversary for the $F{\_max}-OLTSP$ on the real line. A \emph{non-abusive adversary} may only move in a direction if there are yet unserved requests on this side. Our main result is an algorithm which achieves a constant competitive ratio against the non-abusive adversary. T3 - ZIB-Report - 02-36 KW - Online Algorithms KW - Competitive Analysis KW - Comparative Analysis Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7038 ER - TY - GEN A1 - Bley, Andreas A1 - Koch, Thorsten T1 - Integer programming approaches to access and backbone IP-network planning N2 - In this article we study the problem of designing a nation-wide communication network. Such networks usually consist of an access layer, a backbone layer, and maybe several intermediate layers. The nodes of each layer must be connected to those of the next layer in a tree-like fashion. The backbone layer has to satisfy certain survivability and routing constraints. Given the node locations, the demands between them, the possible connections and hardware configurations, and various other technical and administrational constraints, the goal is to decide, which node is assigned to which network level, how the nodes are connected, what hardware must be installed, and how traffic is routed in the backbone. Mixed integer linear programming models and solution methods are presented for both the access and the backbone network design problem. The focus is on the design of IP-over-SDH networks, but the access network design model and large parts of the backbone network design models are general and also applicable for other types of communication networks. Results obtained with these methods in the planning of the German research network are presented. T3 - ZIB-Report - 02-41 KW - Network design KW - Traffic enineering KW - Internet routing KW - Mixed-integer programming Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7081 ER - TY - GEN A1 - Grötschel, Martin A1 - Henk, Martin T1 - On the Representation of Polyhedra by Polynomial Inequalities N2 - A beautiful result of Bröcker and Scheiderer on the stability index of basic closed semi-algebraic sets implies, as a very special case, that every $d$-dimensional polyhedron admits a representation as the set of solutions of at most $d(d+1)/2$ polynomial inequalities. Even in this polyhedral case, however, no constructive proof is known, even if the quadratic upper bound is replaced by any bound depending only on the dimension. Here we give, for simple polytopes, an explicit construction of polynomials describing such a polytope. The number of used polynomials is exponential in the dimension, but in the 2- and 3-dimensional case we get the expected number $d(d+1)/2$. T3 - ZIB-Report - 02-15 KW - polyhedra and polytopes KW - semialgebraic sets KW - polyhedral combinatorics KW - polynomial inequalities Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6826 ER - TY - GEN A1 - Grötschel, Martin T1 - Cardinality Homogeneous Set Systems, Cycles in Matroids, and Associated Polytopes N2 - A subset ${\cal C}$ of the power set of a finite set $E$ is called cardinality homogeneous if, whenever ${\cal C}$ contains some set $F$, ${\cal C}$ contains all subsets of $E$ of cardinality $|F|$. Examples of such set systems ${\cal C}$ are the sets of circuits and the sets of cycles of uniform matroids and the sets of all even or of all odd cardinality subsets of $E$. With each cardinality homogeneous set system ${\cal C}$, we associate the polytope $P({\cal C})$, the convex hull of the incidence vectors of all sets in ${\cal C}$, and provide a complete and nonredundant linear description of $P({\cal C})$. We show that a greedy algorithm optimizes any linear function over $P({\cal C})$, give an explicit optimum solution of the dual linear program, and provide a polynomial time separation algorithm for the class of polytopes of type $P({\cal C})$. T3 - ZIB-Report - 02-19 KW - Cycles in Matroids KW - cardinality homogeneous set systems KW - polytopes KW - greedy algorithm KW - polyhedral combinatorics Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6868 ER - TY - GEN A1 - Krumke, Sven A1 - Poensgen, Diana T1 - Online Call Admission in Optical Networks with Larger Wavelength Demands N2 - In the problem of \emph{Online Call Admission in Optical Networks}, briefly called \textsc{oca}, we are given a graph $G=(V,E)$ together with a set of wavelengths~$W$ and a finite sequence $\sigma=r_1,r_2,\dots$ of calls which arrive in an online fashion. Each call~$r_j$ specifies a pair of nodes to be connected and an integral demand indicating the number of required lightpaths. A lightpath is a path in~$G$ together with a wavelength~$\lambda \in W$. Upon arrival of a call, an online algorithm must decide immediately and irrevocably whether to accept or to reject the call without any knowledge of calls which appear later in the sequence. If the call is accepted, the algorithm must provide the requested number of lightpaths to connect the specified nodes. The essential restriction is the wavelength conflict constraint: each wavelength is available only once per edge, which implies that two lightpaths sharing an edge must have different wavelengths. Each accepted call contributes a benefit equal to its demand to the overall profit. The objective in \textsc{oca} is to maximize the overall profit. Competitive algorithms for \textsc{oca} have been known for the special case where every call requests just a single lightpath. In this paper we present the first competitive online algorithms for the general case of larger demands. T3 - ZIB-Report - 02-22 KW - Call Admission KW - Routing and Wavelength Allocation KW - Optical Networks KW - Competitive Analysis KW - Colorability Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6890 ER - TY - GEN A1 - Krumke, Sven A1 - Marathe, Madhav A1 - Poensgen, Diana A1 - Ravi, Sekharipuram S. A1 - Wirth, Hans-Christoph T1 - Budgeted Maximal Graph Coverage N2 - An instance of the \emph{maximum coverage} problem is given by a set of weighted ground elements and a cost weighted family of subsets of the ground element set. The goal is to select a subfamily of total cost of at most that of a given budget maximizing the weight of the covered elements. We formulate the problem on graphs: In this situation the set of ground elements is specified by the nodes of a graph, while the family of covering sets is restricted to connected subgraphs. We show that on general graphs the problem is polynomial time solvable if restricted to sets of size at most~$2$, but becomes NP-hard if sets of size~$3$ are permitted. On trees, we prove polynomial time solvability if each node appears in a fixed number of sets. In contrast, if vertices are allowed to appear an unbounded number of times, the problem is NP-hard even on stars. We finally give polynomial time algorithms for special cases where the subgraphs form paths and the host graph is a line, a cycle or a star. T3 - ZIB-Report - 02-24 KW - budgeted maximum coverage KW - approximation algorithm KW - dynamic programming Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6918 ER - TY - GEN A1 - Bosse, Hartwig A1 - Grötschel, Martin A1 - Henk, Martin T1 - Polynomial Inequalities Representing Polyhedra N2 - Our main result is that every n-dimensional polytope can be described by at most (2n-1) polynomial inequalities and, moreover, these polynomials can explicitly be constructed. For an n-dimensional pointed polyhedral cone we prove the bound 2n-2 and for arbitrary polyhedra we get a constructible representation by 2n polynomial inequalities. T3 - ZIB-Report - 03-25 KW - polyhedra and polytopes KW - semi-algebraic sets KW - polyhedral combinatorics KW - polynomial inequalities KW - stability index Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7473 ER - TY - GEN A1 - Krumke, Sven T1 - News from the Online Traveling Repairman N2 - The traveling repairman problem (TRP) is a variant of the famous traveling salesman problem (TSP). The objective for the TRP is to minimize the latency, that is the the weighted sum of completion times of the cities, where the completion time of a city is defined to be the time in the tour before the city is reached. In the online traveling repairman problem (OLTRP) requests for visits to cities (points in a metric space) arrive online while the repairman is traveling. We analyze the performance of algorithms using competitive analysis, where the cost of an online algorithm is compared to that of an optimal offline algorithm. An optimal offline algorithm knows the entire request sequence in advance and can serve it with minimum cost. Recently, Feuerstein and Stougie presented a $9$-competitive algorithm for the OLTRP on the real line. In this paper we show how to use techniques from online-scheduling to obtain an $8$-competitive deterministic algorithm which works for any metric space. We also present a randomized algorithm which has a competitive ratio of $\frac{4}{\ln 2}\approx 5.7708$ against an oblivious adversary. All of our results also hold for the ``dial-a-ride'' generalization of the OLTRP, where objects have to be picked up and delivered by a server. T3 - ZIB-Report - 00-08 KW - Traveling Repairman KW - Latency KW - Dial-a-Ride-Problem KW - Competitive Analysis Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5767 ER - TY - GEN A1 - Blom, Michiel A1 - Krumke, Sven A1 - Paepe, Willem de A1 - Stougie, Leen T1 - The Online-TSP Against Fair Adversaries N2 - In the online traveling salesman problem requests for visits to cities (points in a metric space) arrive online while the salesman is traveling. The salesman moves at no more than unit speed and starts and ends his work at a designated origin. The objective is to find a routing for the salesman which finishes as early as possible. Performance of algorithms is measured through their competitive ratio, comparing the outcome of the algorithms with that of an adversary who provides the problem instance and therefore is able to achieve the optimal offline solution. Objections against such omnipotent adversaries have lead us to devise an adversary that is in a natural way, in the context of routing problems, more restricted in power. For the exposition we consider the online traveling salesman problem on the metric space given by the non-negative part of the real line. We show that a very natural strategy is~$3/2$-competitive against the conventional adversary, which matches the lower bound on competitive ratios achievable for algorithms for this problem. Against the more ``\emph{fair adversary}'', that we propose, we show that there exists an algorithm with competitive ratio $\frac{1+\sqrt{17}}{4}\approx 1.28$ and provide a matching lower bound. We also show competitiveness results for a special class of algorithms (called zealous algorithms) that do not allow waiting time for the server as long as there are requests unserved. T3 - ZIB-Report - 00-09 KW - Vehicle Routing KW - Online-Algorithms KW - Competitive Analysis Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5779 ER - TY - GEN A1 - Grötschel, Martin A1 - Krumke, Sven A1 - Rambau, Jörg T1 - Wo bleibt der Aufzug? N2 - Dieser Artikel gibt eine allgemeinverständliche Einführung in die spezielle Problematik kombinatorischer Online-Problem am Beispiel der Fahrstuhlsteuerung. T3 - ZIB-Report - SC-99-29 KW - Aufzugsteuerung KW - Online-Optimierung KW - Echtzeit-Optimierung KW - kompetitive Analyse Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4175 ER - TY - GEN A1 - Ascheuer, Norbert A1 - Fischetti, Matteo A1 - Grötschel, Martin T1 - Solving the Asymmetric Travelling Salesman Problem with Time Windows by Branch-and-Cut N2 - Many optimization problems have several equivalent mathematical models. It is often not apparent which of these models is most suitable for practical computation, in particular, when a certain application with a specific range of instance sizes is in focus. Our paper addresses the Asymmetric Travelling Salesman Problem with time windows (ATSP-TW) from such a point of view. The real--world application we aim at is the control of a stacker crane in a warehouse. We have implemented codes based on three alternative integer programming formulations of the ATSP-TW and more than ten heuristics. Computational results for real-world instances with up to 233 nodes are reported, showing that a new model presented in a companion paper outperforms the other two models we considered --- at least for our special application --- and that the heuristics provide acceptable solutions. T3 - ZIB-Report - SC-99-31 KW - Asymmetric Travelling Salesman Problem KW - Time Windows KW - Integer Programs KW - Branch&Cut-Algorithm KW - Heuristics KW - Control of Stacker Cranes Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4191 ER - TY - GEN A1 - Hauptmeier, Dietrich A1 - Krumke, Sven A1 - Rambau, Jörg T1 - The Online Dial-a-Ride Problem under Reasonable Load N2 - In this paper, we analyze algorithms for the online dial-a-ride problem with request sets that fulfill a certain worst-case restriction: roughly speaking, a set of requests for the online dial-a-ride problem is reasonable if the requests that come up in a sufficiently large time period can be served in a time period of at most the same length. This new notion is a stability criterion implying that the system is not overloaded. The new concept is used to analyze the online dial-a-ride problem for the minimization of the maximal resp.\ average flow time. Under reasonable load it is possible to distinguish the performance of two particular algorithms for this problem, which seems to be impossible by means of classical competitive analysis. T3 - ZIB-Report - SC-99-08 KW - online optimization KW - competitive analysis KW - elevator Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3961 ER - TY - GEN A1 - Grötschel, Martin A1 - Hauptmeier, Dietrich A1 - Krumke, Sven A1 - Rambau, Jörg T1 - Simulation Studies for the Online-Dial-a-Ride Problem N2 - In a large distribution center of Herlitz AG, Berlin, we invesigated the elevator subsystem of the fully automated pallet transportation system. Each elevator may carry one pallet and has to serve eight levels. The goal is to minimize the average resp.\ the maximum flow time. The variants of this elevator control problem have been subject of recent theoretical research and are known as online-dial-a-ride problems. In this paper we investigate several online algorithms for several versions of online-dial-a-ride problems by means of a simulation program, developed on the basis of the simulation library AMSEL. We draw statistics from samples of randomly generated data providing for different load situations. Moreover, we provide preliminary studies with real production data for a system of five elevators connected by a conveyor circuit, as can be found at the Herlitz plant. We show which algorithms are best under certain load situations and which lead to break downs under particular circumstances. T3 - ZIB-Report - SC-99-09 KW - online optimization KW - competitive analysis KW - elevator KW - simulation studies Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3976 ER - TY - THES A1 - Koch, Thorsten T1 - Rapid Mathematical Programming N2 - The thesis deals with the implementation and application of out-of-the-box tools in linear and mixed integer programming. It documents the lessons learned and conclusions drawn from five years of implementing, maintaining, extending, and using several computer codes to solve real-life industrial problems. By means of several examples it is demonstrated how to apply algebraic modeling languages to rapidly devise mathematical models of real-world problems. It is shown that today's MIP solvers are capable of solving the resulting mixed integer programs, leading to an approach that delivers results very quickly. Even though, problems are tackled that not long ago required the implementation of specialized branch-and-cut algorithms. In the first part of the thesis the modeling language Zimpl is introduced. Chapter 2 contains a complete description of the language. In the subsequent chapter details of the implementation are described. Both theoretical and practical considerations are discussed. Aspects of software engineering, error prevention, and detection are addressed. In the second part several real-world projects are examined that employed the methodology and the tools developed in the first part. Chapter 4 presents three projects from the telecommunication industry dealing with facility location problems. Chapter 5 characterizes questions that arise in UMTS planning. Problems, models, and solutions are discussed. Special emphasis is put on the dependency of the precision of the input data and the results. Possible reasons for unexpected and undesirable solutions are explained. Finally, the Steiner tree packing problem in graphs, a well-known hard combinatorial problem, is revisited. A formerly known, but not yet used model is applied to combine switchbox wire routing and via minimization. All instances known from the literature are solved by this approach, as are some newly generated bigger problem instances. T3 - ZIB-Report - 04-58 KW - Modelling Languages KW - Mixed Integer Programming KW - Steiner Tree Packing in Graphs KW - Location Planning in Telecommunications KW - UMTS Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8346 ER - TY - GEN A1 - Bley, Andreas T1 - Inapproximability Results for the Inverse Shortest Paths Problem with Integer Length and Unique Shortest Paths N2 - We study the complexity of two Inverse Shortest Paths (ISP) problems with integer arc lengths and the requirement for uniquely determined shortest paths. Given a collection of paths in a directed graph, the task is to find positive integer arc lengths such that the given paths are uniquely determined shortest paths between their respective terminals. The first problem seeks for arc lengths that minimize the length of the longest of the prescribed paths. In the second problem, the length of the longest arc is to be minimized. We show that it is $np-hard$ to approximate the minimal longest path length within a factor less than $8/7$ or the minimal longest arc length within a factor less than $9/8$. This answers the (previously) open question whether these problems are $np-hard$ or not. We also present a simple algorithm that achieves an $\mathcal{O}(|V|)$-approximation guarantee for both variants. Both ISP problems arise in the planning of telecommunication networks with shortest path routing protocols. Our results imply that it is $\mathcal{NP}$-hard to decide whether a given path set can be realized with a real shortest path routing protocol such as OSPF, IS-IS, or RIP. T3 - ZIB-Report - 05-04 KW - Inverse Shortest Paths KW - Computational Complexity KW - Approximation KW - Shortest Path Routing Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8388 ER - TY - GEN A1 - Koster, Arie M.C.A. T1 - Wavelength Assignment in Multi-Fiber WDM Networks by Generalized Edge Coloring N2 - In this paper, we study wavelength assignment problems in multi-fiber WDM networks. We focus on the special case that all lightpaths have at most two links. This in particular holds in case the network topology is a star. As the links incident to a specific node in a meshed topology form a star subnetwork, results for stars are also of interest for general meshed topologies. We show that wavelength assignment with at most two links per lightpath can be modeled as a generalized edge coloring problem. By this relation, we show that for a network with an even number of fibers at all links and at most two links per lightpath, all lightpaths can be assigned a wavelength without conversion. Moreover, we derive a lower bound on the number of lightpaths to be converted for networks with arbitrary numbers of fibers at the links. A comparison with linear programming lower bounds reveals that the bounds coincide for problems with at most two links per lightpath. For meshed topologies, the cumulative lower bound over all star subnetworks equals the best known solution value for all realistic wavelength assignment instances available, by this proving optimality. T3 - ZIB-Report - 05-13 KW - wavelength assignment KW - optical networks KW - graph theory KW - combinatorial optimization KW - integer programming Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8478 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Grötschel, Martin A1 - Pfetsch, Marc T1 - Models for Line Planning in Public Transport N2 - The \emph{line planning problem} is one of the fundamental problems in strategic planning of public and rail transport. It consists of finding lines and corresponding frequencies in a public transport network such that a given travel demand can be satisfied. There are (at least) two objectives. The transport company wishes to minimize its operating cost; the passengers request short travel times. We propose two new multi-commodity flow models for line planning. Their main features, in comparison to existing models, are that the passenger paths can be freely routed and that the lines are generated dynamically. T3 - ZIB-Report - 04-10 KW - line planning KW - column generation Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7854 ER - TY - GEN A1 - Grötschel, Martin T1 - My Favorite Theorem: Characterizations of Perfect Graphs N2 - This paper summarizes and discusses various characterizations of perfect graphs and mentions some open problems in this area. T3 - ZIB-Report - SC-99-17 KW - graph theory KW - perfect graphs KW - integer programming Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4053 ER - TY - GEN A1 - Bienstock, Daniel A1 - Bley, Andreas T1 - Capacitated Network Design with Multicast Commodities N2 - This paper addresses the problem of designing a minimum cost network whose capacities are sufficiently large to allow a feasible routing of a given set of multicast commodities. A multicast commodity consists of a set of two or mo re terminals that need to be connected by a so called broadcast tree, which consumes on all of its edges a capacity as large as the demand value associated with that commodity. We model the network design problem with multicast commodities as the problem of packing capacitated Steiner trees in a graph. In the first part of the paper we present three mixed-integer programming formulations for this problem. The first natural formulation uses only one integer capacity variable for each edge and and one binary tree variable for each commodity-edge pair. Applying well-known techniques from the Steiner tree problem, we then develop a stronger directed and a multicommodity flow based mixed-integer programming formulation. In the second part of the paper we study the associated polyhedra and derive valid and even facet defining inequalities for the natural formulation. Finally, we describe separation algorithms for these inequalities and present computational results that demonstrate the strength of our extended formulations. T3 - ZIB-Report - 00-14 KW - multicast network design KW - Steiner tree packing Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5823 ER - TY - GEN A1 - Krumke, Sven A1 - Noltemeier, Hartmut A1 - Wirth, Hans-Christoph T1 - Graphentheoretische Konzepte und Algorithmen N2 - Das vorliegende Skript bietet eine Einf{ü}hrung in die Graphentheorie und graphentheoretische Algorithmen. Im zweiten Kapitel werden Grundbegriffe der Graphentheorie vorgestellt. Das dritte Kapitel besch{ä}ftigt sich mit der Existenz von Wegen in Graphen. Hier wird auch die L{ö}suung des ber{ü}hmten K{ö}nigsberger Br{ü}ckenproblems aufgezeigt und der Satz von Euler bewiesen. Im vierten Kapitel wird gezeigt, wie man auf einfache Weise die Zusammenhangskomponenten eines Graphen bestimmen kann. Im Kapitel sechs wird dann sp{ä}ter mit der Tiefensuche ein Verfahren vorgestellt, das schneller arbeitet und mit dessen Hilfe man noch mehr Informationen {ü}ber die Struktur eines Graphen gewinnen kann. In den folgenden Kapiteln werden Algorithmen vorgestellt, um minimale aufspannenden B{ä}ume, k{ü}rzeste Wege und maximale Fl{ü}sse in Graphen zu bestimmen. Am Ende des Skripts wird ein kurzer Einblick in die planaren Graphen und Graphhomomorphismen geboten. T3 - ZIB-Report - 00-19 KW - Graphen KW - Algorithmen KW - Komplexit{ä}t KW - kombinatorische Optimierung Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5873 ER - TY - GEN A1 - Hoesel, Stan P.M. van A1 - Koster, Arie M.C.A. A1 - Leensel, Robert L.M.J. van de A1 - Savelsbergh, Martin W.P. T1 - Polyhedral Results for the Edge Capacity Polytope N2 - 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. T3 - ZIB-Report - 00-22 KW - network design KW - mixed integer programming KW - cutting planes KW - knapsack with integer capacity Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5907 ER - TY - GEN A1 - Krumke, Sven A1 - Rambau, Jörg T1 - Online Optimierung N2 - Wie soll man einen Aufzug steuern, wenn man keine Informationen über zukünftige Fahraufträge besitzt? Soll man eine Bahncard kaufen, wenn die nächsten Bahnreisen noch unbekannt sind? In der klassischen kombinatorischen Optimierung geht man davon aus, daß die Daten jeder Probleminstanz vollständig gegeben sind. In vielen Fällen modelliert diese \emph{Offline-Optimierung} jedoch die Situationen aus Anwendungen nur ungenügend. Zahlreiche Problemstellungen in der Praxis sind in natürlicher Weise \emph{online}: Sie erfordern Entscheidungen, die unmittelbar und ohne Wissen zukünftiger Ereignisse getroffen werden müssen. Als ein Standardmittel zur Beurteilung von Online-Algorithmen hat sich die \emph{kompetitive Analyse} durchgesetzt. Dabei vergleicht man den Zielfunktionswert einer vom Online-Algorithmus generierten Lösung mit dem Wert einer optimalen Offline-Lösung. Mit Hilfe der kompetitiven Analyse werden im Skript Algorithmen zum Caching, Netzwerk-Routing, Scheduling und zu Transportaufgaben untersucht. Auch die Schwächen der kompetitiven Analyse werden aufgezeigt und alternative Analysekonzepte vorgestellt. Neben der theoretischen Seite werden auch die Anwendungen der Online-Optimierung in der Praxis, vor allem bei Problemen der innerbetrieblichen Logistik, beleuchtet. Bei der Steuerung automatischer Transportsysteme tritt eine Fülle von Online-Problemen auf. Hierbei werden an die Algorithmen oftmals weitere Anforderungen gestellt. So müssen Entscheidungen unter strikten Zeitbeschränkungen gefällt werden (Echtzeit-Anforderungen). Dieses Skript ist aus dem Online-Teil der Vorlesung -Ausgewählte Kapitel aus der ganzzahligen Optimierung- (Wintersemester~1999/2000) und der Vorlesung -Online Optimierung- (Sommersemester~2000) an der Technischen Universität Berlin entstanden. T3 - ZIB-Report - 00-55 KW - Kompetitive Analyse KW - Online Optimierung KW - Online Algorithmen Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6238 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Grötschel, Martin A1 - Löbel, Andreas T1 - Duty Scheduling in Public Transit N2 - This article is about \emph{adaptive column generation techniques} for the solution of duty scheduling problems in public transit. The current optimization status is exploited in an adaptive approach to guide the subroutines for duty generation, LP resolution, and schedule construction toward relevant parts of a large problem. Computational results for three European scenarios are reported. T3 - ZIB-Report - 01-02 KW - Duty Scheduling KW - Constraint Shortest Paths Problem KW - Column Generation Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6286 ER - TY - GEN A1 - Grötschel, Martin A1 - Krumke, Sven A1 - Rambau, Jörg A1 - Winter, Thomas A1 - Zimmermann, Uwe T1 - Combinatorial Online Optimization in Real Time N2 - Optimization is the task of finding an optimum solution to a given problem. When the decision variables are discrete we speak of a combinatorial optimization problem. Such a problem is online when decisions have to be made before all data of the problem are known. And we speak of a real-time online problem when online decisions have to be computed within very tight time bounds. This paper surveys the are of combinatorial online and real-time optimization, it discusses, in particular, the concepts with which online and real-time algorithms can be analyzed. T3 - ZIB-Report - 01-16 KW - Online Optimization KW - Realtime Optimization KW - Competitive Analysis KW - Heuristics KW - Survey Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6424 ER - TY - GEN A1 - Helmberg, Christoph T1 - A Cutting Plane Algorithm for Large Scale Semidefinite Relaxations N2 - The recent spectral bundle method allows to compute, within reasonable time, approximate dual solutions of large scale semidefinite quadratic 0-1 programming relaxations. We show that it also generates a sequence of primal approximations that converge to a primal optimal solution. Separating with respect to these approximations gives rise to a cutting plane algorithm that converges to the optimal solution under reasonable assumptions on the separation oracle and the feasible set. We have implemented a practical variant of the cutting plane algorithm for improving semidefinite relaxations of constrained quadratic 0-1 programming problems by odd-cycle inequalities. We also consider separating odd-cycle inequalities with respect to a larger support than given by the cost matrix and present a heuristic for selecting this support. Our preliminary computational results for max-cut instances on toroidal grid graphs and balanced bisection instances indicate that warm start is highly efficient and that enlarging the support may sometimes improve the quality of relaxations considerably. T3 - ZIB-Report - 01-26 KW - bisection KW - equicut KW - max-cut KW - semidefinite programming KW - spectral bundle method KW - subgradient method KW - quadratic 0-1 programming Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6527 ER - TY - GEN A1 - Krumke, Sven A1 - Rambau, Jörg T1 - Probieren geht über Studieren? Entscheidungshilfen für kombinatorische Online-Optimierungsprobleme in der innerbetrieblichen Logistik N2 - Die Automatisierung von innerbetrieblicher Logistik erfordert -- über die physikalische Steuerung von Geräten hinaus -- auch eine effiziente Organisation der Transporte: ein Aufgabenfeld der kombinatorischen Optimierung. Dieser Artikel illustriert anhand von konkreten Aufgabenstellungen die Online-Problematik (unvollständiges Wissen) sowie die Echtzeit-Problematik (beschränkte Rechenzeit), auf die man in der innerbetrieblichen Logistik trifft. Der Text gibt einen Überblick über allgemeine Konstruktionsprinzipien für Online-Algorithmen und Bewertungsmethoden, die bei der Entscheidung helfen, welche Algorithmen für eine vorliegende Problemstellung geeignet sind. T3 - ZIB-Report - 02-05 KW - Logistik KW - Hochregallagerbediengeräte KW - Kommissioniermobile KW - Aufzüge KW - Online-Optimierung KW - Echtzeit-Optimierung KW - Simulation KW - kompetitive Analyse Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6723 ER - TY - GEN A1 - Pfetsch, Marc T1 - A Branch-And-Cut for the Maximum Feasible Subsystem Problem N2 - We present a branch-and-cut algorithm for the NP-hard maximum feasible subsystem problem: For a given infeasible linear inequality system, determine a feasible subsystem containing as many inequalities as possible. The complementary problem, where one has to remove as few inequalities as possible in order to render the system feasible, can be formulated as a set covering problem. The rows of this formulation correspond to irreducible infeasible subsystems, which can be exponentially many. The main issue of a branch-and-cut algorithm for MaxFS is to efficiently find such infeasible subsystems. We present three heuristics for the corresponding NP-hard separation problem and discuss further cutting planes. This paper contains an extensive computational study of our implementation on a variety of instances arising in a number of applications. T3 - ZIB-Report - 05-46 KW - infeasible linear inequality system KW - irreducible infeasible subsystem KW - (IIS) maximum feasible subsystem problem KW - minimum IIS-cover KW - branch-and-cut Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8797 ER - TY - GEN A1 - Kaibel, Volker A1 - Koch, Thorsten T1 - Mathematik für den Volkssport N2 - "`Volkssport Sudoku"' titelt der Stern in seiner Ausgabe vom 24. Mai2006. In der Tat traut sich derzeit kaum noch eine Zeitung, ohne Sudoku zu erscheinen. Die Begeisterung am Lösen dieser Zahlenrätsel offenbart eine unvermutete Freude am algorithmischen Arbeiten. Mathematisch kann man Sudokus als lineare diophantische Gleichungssysteme mit Nichtnegativitätsbedingungen formulieren. Solche ganzzahligen linearen Programme sind die wichtigsten Modellierungswerkzeuge in zahlreichen Anwendungsgebieten wie z.B. der Optimierung von Telekommunikations- und Verkehrsnetzen. Moderne Verfahren zur Lösung dieser Optimierungsprobleme sind durch Sudokus allerdings deutlich weniger zu beeindrucken als Zeitungsleser. T3 - ZIB-Report - 06-28 KW - Sudoku KW - Integer Programming KW - General Public Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9225 ER -