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 - 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 - THES A1 - Krumke, Sven T1 - Online Optimization: Competitive Analysis and Beyond N2 - Traditional optimization techniques assume, in general, knowledge of all data of a problem instance. There are many cases in practice, however, where decisions have to be made before complete information about the data is available. In fact, it may be necessary to produce a part of the problem solution as soon as a new piece of information becomes known. This is called an \emph{online situation}, and an algorithm is termed \emph{online}, if it makes a decision (computes a partial solution) whenever a new piece of data requests an action. \emph{Competitive analysis} has become a standard yardstick to measure the quality of online algorithms. One compares the solution produced by an online algorithm to that of an optimal (clairvoyant) offline algorithm. An online algorithm is called $c$-competitive if on every input the solution it produces has cost'' at most $c$~times that of the optimal offline algorithm. This situation can be imagined as a game between an online player and a malicious adversary. Although competitive analysis is a worst-case analysis and henceforth pessimistic, it often allows important insights into the problem structure. One can obtain an idea about what kind of strategies are promising for real-world systems and why. On the other hand there are also cases where the offline adversary is simply too powerful and allows only trivial competitiveness results. This phenomenon is called hitting the triviality barrier''. We investigate several online problems by means of competitive analysis. We also introduce new concepts to overcome the weaknesses of the standard approach and to go beyond the triviality barrier. T3 - ZIB-Report - 02-25 KW - competitive analysis KW - online optimization KW - online algorithm KW - approximation algorithm Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6925 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 - 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 - Lukac, Sascha A1 - Mitusch, Kay A1 - Schlechte, Thomas A1 - Schultz, Sören A1 - Tanner, Andreas T1 - An Auctioning Approach to Railway Slot Allocation N2 - We present an approach to implement an auction of railway slots. Railway network, train driving characteristics, and safety requirements are described by a simplified, but still complex macroscopic model. In this environment, slots are modelled as combinations of scheduled track segments. The auction design builds on the iterative combinatorial auction. However, combinatorial bids are restricted to some types of slot bundles that realize positive synergies between slots. We present a bidding language that allows bidding for these slot bundles. An integer programming approach is proposed to solve the winner determination problem of our auction. Computational results for auction simulations in the Hannover-Fulda-Kassel area of the German railway network give evidence that auction approaches can induce a more efficient use of railway capacity. T3 - ZIB-Report - 05-45 KW - Railway Slot Allocation KW - Train Dispatching KW - Combinatorial Auctions Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8786 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 - 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 - 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 - 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 - 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 - Borndörfer, Ralf T1 - Combinatorial Packing Problems N2 - This article investigates a certain class of combinatorial packing problems and some polyhedral relations between such problems and the set packing problem. T3 - ZIB-Report - 03-03 KW - Packing problems KW - polyhedral combinatorics Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7255 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 - 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 - Borndörfer, Ralf A1 - Neumann, Marika A1 - Pfetsch, Marc T1 - The Steiner Connectivity Problem N2 - The Steiner connectivity problem is a generalization of the Steiner tree problem. It consists in finding a minimum cost set of simple paths to connect a subset of nodes in an undirected graph. We show that polyhedral and algorithmic results on the Steiner tree problem carry over to the Steiner connectivity problem, namely, the Steiner cut and the Steiner partition inequalities, as well as the associated polynomial time separation algorithms, can be generalized. Similar to the Steiner tree case, a directed formulation, which is stronger than the natural undirected one, plays a central role. T3 - ZIB-Report - 09-07 KW - Steiner Tree KW - Partition Inequalities KW - Paths KW - Connectivity KW - Generalization Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11171 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Cardonha, Carlos T1 - A Set Partitioning Approach to Shunting N2 - The Vehicle Positioning Problem (VPP) is a classical combinatorial optimization problem in public transport planning. A number of models and approaches have been suggested in the literature, which work for small problems, but not for large ones. We propose in this article a novel set partitioning model and an associated column generation solution approach for the VPP. The model provides a tight linear description of the problem. The pricing problem, and hence the LP relaxation itself, can be solved in polynomial resp. pseudo-polynomial time for some versions of the problems. T3 - ZIB-Report - 09-18 KW - integer programming KW - set partitioning KW - combinatorial packing KW - vehicle positioning Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11326 SN - 1438-0064 ER - TY - GEN A1 - Berthold, Timo A1 - Heinz, Stefan A1 - Vigerske, Stefan T1 - Extending a CIP framework to solve MIQCPs N2 - This paper discusses how to build a solver for mixed integer quadratically constrained programs (MIQCPs) by extending a framework for constraint integer programming (CIP). The advantage of this approach is that we can utilize the full power of advanced MIP and CP technologies. In particular, this addresses the linear relaxation and the discrete components of the problem. For relaxation, we use an outer approximation generated by linearization of convex constraints and linear underestimation of nonconvex constraints. Further, we give an overview of the reformulation, separation, and propagation techniques that are used to handle the quadratic constraints efficiently. We implemented these methods in the branch-cut-and-price framework SCIP. Computational experiments indicates the potential of the approach. T3 - ZIB-Report - 09-23 KW - mixed integer quadratically constrained programming KW - constraint integer programming KW - convex relaxation KW - nonconvex Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11371 SN - 1438-0064 ER - TY - GEN A1 - Achterberg, Tobias A1 - Berthold, Timo A1 - Koch, Thorsten A1 - Wolter, Kati T1 - Constraint Integer Programming: a New Approach to Integrate CP and MIP N2 - This article introduces constraint integer programming (CIP), which is a novel way to combine constraint programming (CP) and mixed integer programming (MIP) methodologies. CIP is a generalization of MIP that supports the notion of general constraints as in CP. This approach is supported by the CIP framework SCIP, which also integrates techniques from SAT solving. SCIP is available in source code and free for non-commercial use. We demonstrate the usefulness of CIP on two tasks. First, we apply the constraint integer programming approach to pure mixed integer programs. Computational experiments show that SCIP is almost competitive to current state-of-the-art commercial MIP solvers. Second, we employ the CIP framework to solve chip design verification problems, which involve some highly non-linear constraint types that are very hard to handle by pure MIP solvers. The CIP approach is very effective here: it can apply the full sophisticated MIP machinery to the linear part of the problem, while dealing with the non-linear constraints by employing constraint programming techniques. T3 - ZIB-Report - 08-01 KW - Constraint Programming KW - Ganzzahlige Programmierung KW - Branch-And-Cut KW - Optimierungssoftware KW - Chipverifikation KW - constraint programming KW - mixed integer programming KW - branch-and-cut KW - optimization software KW - chip verification Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10520 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Neumann, Marika A1 - Pfetsch, Marc T1 - Angebotsplanung im öffentlichen Nahverkehr T1 - Service Design in Public Transport N2 - Die Angebotsplanung im öffentlichen Nahverkehr umfasst die Aufgaben der Netz-, Linien-,Fahr- und Preisplanung. Wir stellen zwei mathematische Optimierungsmodelle zur Linien- und Preisplanung vor. Wir zeigen anhand von Berechnungen für die Verkehrsbetriebe in Potsdam(ViP), dass sich damit komplexe Zusammenhänge quantitativ analysieren lassen. Auf diese Weise untersuchen wir die Auswirkungen von Freiheitsgraden auf die Konstruktion von Linien und die Wahl von Reisewegen der Passagiere, Abhängigkeiten zwischen Kosten und Reisezeiten sowie den Einfluss verschiedener Preissysteme auf Nachfrage und Kostendeckung. T3 - ZIB-Report - 08-04 KW - Preisplanung KW - Linienplanung KW - Optimierung Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10555 SN - 1438-0064 ER - TY - GEN A1 - Bley, Andreas T1 - On the Hardness of Finding Small Shortest Path Routing Conflicts N2 - Nowadays most data networks use shortest path protocols such as OSPF or IS-IS to route traffic. Given administrative routing lengths for the links of a network, all data packets are sent along shortest paths with respect to these lengths from their source to their destination. One of the most fundamental problems in planning shortest path networks is to decide whether a given set of routing paths forms a valid routing and, if this is not the case, to find a small subset of the given paths that cannot be shortest paths simultaneously for any routing lengths. In this paper we show that it is NP-hard to approximate the size of the smallest shortest path conflict by a factor less than 7/6. T3 - ZIB-Report - 09-15 KW - shortest path routing KW - computational complexity Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11276 SN - 1438-0064 ER - TY - GEN A1 - Erol, Berkan A1 - Klemenz, Marc A1 - Schlechte, Thomas A1 - Schultz, Sören A1 - Tanner, Andreas T1 - TTPLIB 2008 - A Library for Train Timetabling Problems N2 - We introduce (TTPlib), a data library for train timetabling problems that can be accessed at http://ttplib.zib.de. In version 1.0, the library contains data related to 50 scenarios. Most instances result from the combination of macroscopic railway networks and several train request sets for the German long distance area containing Hannover, Kassel and Fulda, short denoted by Ha-Ka-Fu. In this paper, we introduce the data concepts of TTPlib, describe the scenarios included in the library and provide a free visualization tool TraVis. T3 - ZIB-Report - 08-19 KW - Fahrplanung KW - Problembibliothek KW - Optimierung KW - Train Timetabling Problem KW - Data Library KW - Optimization Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10732 SN - 1438-0064 ER - TY - GEN A1 - Schlechte, Thomas A1 - Borndörfer, Ralf T1 - Balancing Efficiency and Robustness – A Bi-criteria Optimization Approach to Railway Track Allocation N2 - Technical restrictions and challenging details let railway traffic become one of the most complex transportation systems. Routing trains in a conflict-free way through a track network is one of the basic scheduling problems for any railway company. This article focuses on a robust extension of this problem, also known as train timetabling problem (TTP), which consists in finding a schedule, a conflict free set of train routes, of maximum value for a given railway network. However, timetables are not only required to be profitable. Railway companies are also interested in reliable and robust solutions. Intuitively, we expect a more robust track allocation to be one where disruptions arising from delays are less likely to be propagated causing delays of subsequent trains. This trade-off between an efficient use of railway infrastructure and the prospects of recovery leads us to a bi-criteria optimization approach. On the one hand we want to maximize the profit of a schedule, that is more or less to maximize the number of feasible routed trains. On the other hand if two trains are scheduled as tight as possible after each other it is clear that a delay of the first one always affects the subsequent train. We present extensions of the integer programming formulation in [BorndoerferSchlechte2007] for solving (TTP). These models can incorporate both aspects, because of the additional track configuration variables. We discuss how these variables can directly be used to measure a certain type of robustness of a timetable. For these models which can be solved by column generation techniques, we propose so-called scalarization techniques, see [Ehrgott2005], to determine efficient solutions. Here, an efficient solution is one which does not allow any improvement in profit and robustness at the same time. We prove that the LP-relaxation of the (TTP) including an additional $\epsilon$-constraint remains solvable in polynomial time. Finally, we present some preliminary results on macroscopic real-world data of a part of the German long distance railway network. T3 - ZIB-Report - 08-22 KW - Train Timetabling Problem KW - Bicriteria Optimization KW - Column Generation Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10763 SN - 1438-0064 ER - TY - GEN A1 - Bley, Andreas T1 - An Integer Programming Algorithm for Routing Optimization in IP Networks N2 - Most data networks nowadays use shortest path protocols to route the traffic. Given administrative routing lengths for the links of the network, all data packets are sent along shortest paths with respect to these lengths from their source to their destination. In this paper, we present an integer programming algorithm for the minimum congestion unsplittable shortest path routing problem, which arises in the operational planning of such networks. Given a capacitated directed graph and a set of communication demands, the goal is to find routing lengths that define a unique shortest path for each demand and minimize the maximum congestion over all links in the resulting routing. We illustrate the general decomposition approach our algorithm is based on, present the integer and linear programming models used to solve the master and the client problem, and discuss the most important implementational aspects. Finally, we report computational results for various benchmark problems, which demonstrate the efficiency of our algorithm. T3 - ZIB-Report - 08-30 KW - Shortest Path Routing KW - Integer Programming Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10814 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Neumann, Marika A1 - Pfetsch, Marc T1 - The Line Connectivity Problem N2 - This paper introduces the "line connectivity problem", a generalization of the Steiner tree problem and a special case of the line planning problem. We study its complexity and give an IP formulation in terms of an exponential number of constraints associated with "line cut constraints". These inequalities can be separated in polynomial time. We also generalize the Steiner partition inequalities. T3 - ZIB-Report - 08-31 KW - Steiner Tree Generalization Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10820 SN - 1438-0064 ER - TY - GEN A1 - Torres, Luis Miguel A1 - Torres, Ramiro A1 - Borndörfer, Ralf A1 - Pfetsch, Marc T1 - Line Planning on Paths and Tree Networks with Applications to the Quito Trolebus System N2 - Line planning is an important step in the strategic planning process of a public transportation system. In this paper, we discuss an optimization model for this problem in order to minimize operation costs while guaranteeing a certain level of quality of service, in terms of available transport capacity. We analyze the problem for path and tree network topologies as well as several categories of line operation that are important for the Quito Trolebus system. It turns out that, from a computational complexity worst case point of view, the problem is hard in all but the most simple variants. In practice, however, instances based on real data from the Trolebus System in Quito can be solved quite well, and significant optimization potentials can be demonstrated. T3 - ZIB-Report - 08-35 KW - line planning KW - computational complexity KW - public transport KW - integer programming Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10869 SN - 1438-0064 ER - TY - GEN A1 - Stephan, Rüdiger T1 - Cardinality Constrained Combinatorial Optimization: Complexity and Polyhedra N2 - Given a combinatorial optimization problem and a subset $N$ of natural numbers, we obtain a cardinality constrained version of this problem by permitting only those feasible solutions whose cardinalities are elements of $N$. In this paper we briefly touch on questions that addresses common grounds and differences of the complexity of a combinatorial optimization problem and its cardinality constrained version. Afterwards we focus on polytopes associated with cardinality constrained combinatorial optimization problems. Given an integer programming formulation for a combinatorial optimization problem, by essentially adding Grötschel's cardinality forcing inequalities, we obtain an integer programming formulation for its cardinality restricted version. Since the cardinality forcing inequalities in their original form are mostly not facet defining for the associated polyhedra, we discuss possibilities to strengthen them. T3 - ZIB-Report - 08-48 KW - Combinatorial Optimization KW - Cardinality Constraints KW - Matroids KW - Cardinality Forcing Inequalities Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11026 SN - 1438-0064 ER - TY - GEN A1 - Torres, Luis Miguel A1 - Torres, Ramiro A1 - Borndörfer, Ralf A1 - Pfetsch, Marc T1 - On the Line Planning Problem in Tree Networks N2 - We introduce an optimization model for the line planning problem in a public transportation system that aims at minimizing operational costs while ensuring a given level of quality of service in terms of available transport capacity. We discuss the computational complexity of the model for tree network topologies and line structures that arise in a real-world application at the Trolebus Integrated System in Quito. Computational results for this system are reported. T3 - ZIB-Report - 08-52 KW - line planning KW - computational complexity KW - public transport optimization Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11069 SN - 1438-0064 ER - TY - GEN A1 - Torres, Luis Miguel A1 - Torres, Ramiro A1 - Borndörfer, Ralf A1 - Pfetsch, Marc T1 - Line Planning on Paths and Tree Networks with Applications to the Quito Trolebus System (Extended Abstract) N2 - Line planning is an important step in the strategic planning process of a public transportation system. In this paper, we discuss an optimization model for this problem in order to minimize operation costs while guaranteeing a certain level of quality of service, in terms of available transport capacity. We analyze the problem for path and tree network topologies as well as several categories of line operation that are important for the Quito Trolebus system. It turns out that, from a computational complexity worst case point of view, the problem is hard in all but the most simple variants. In practice, however, instances based on real data from the Trolebus System in Quito can be solved quite well, and significant optimization potentials can be demonstrated. T3 - ZIB-Report - 08-53 KW - line planning KW - computational complexity KW - public transport KW - integer programming Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11076 SN - 1438-0064 ER - TY - GEN A1 - Stephan, Rüdiger T1 - On the cardinality constrained matroid polytope N2 - Edmonds showed that the so-called rank inequalities and the nonnegativity constraints provide a complete linear description of the matroid polytope. By essentially adding Grötschel's cardinality forcing inequalities, we obtain a complete linear description of the cardinality constrained matroid polytope which is the convex hull of the incidence vectors of those independent sets that have a feasible cardinality. Moreover, we show how the separation problem for the cardinality forcing inequalities can be reduced to that for the rank inequalities. We also give necessary and sufficient conditions for a cardinality forcing inequality to be facet defining. T3 - ZIB-Report - 08-08 KW - Matroid-Polytop KW - Kardinalitätsbeschränkung KW - Separationsalgorithmus KW - Matroid Polytope KW - Cardinality constraints KW - Separation algorithm Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10614 SN - 1438-0064 ER - TY - GEN A1 - Berthold, Timo A1 - Heinz, Stefan A1 - Pfetsch, Marc T1 - Solving Pseudo-Boolean Problems with SCIP N2 - Pseudo-Boolean problems generalize SAT problems by allowing linear constraints and a linear objective function. Different solvers, mainly having their roots in the SAT domain, have been proposed and compared,for instance, in Pseudo-Boolean evaluations. One can also formulate Pseudo-Boolean models as integer programming models. That is,Pseudo-Boolean problems lie on the border between the SAT domain and the integer programming field. In this paper, we approach Pseudo-Boolean problems from the integer programming side. We introduce the framework SCIP that implements constraint integer programming techniques. It integrates methods from constraint programming, integer programming, and SAT-solving: the solution of linear programming relaxations, propagation of linear as well as nonlinear constraints, and conflict analysis. We argue that this approach is suitable for Pseudo-Boolean instances containing general linear constraints, while it is less efficient for pure SAT problems. We present extensive computational experiments on the test set used for the Pseudo-Boolean evaluation 2007. We show that our approach is very efficient for optimization instances and competitive for feasibility problems. For the nonlinear parts, we also investigate the influence of linear programming relaxations and propagation methods on the performance. It turns out that both techniques are helpful for obtaining an efficient solution method. T3 - ZIB-Report - 08-12 KW - Pseudo-Boolean KW - Constraint Programming KW - Ganzzahlige Programmierung KW - Branch-And-Cut KW - Optimierungssoftware KW - Pseudo-Boolean KW - constraint integer programming KW - integer programming KW - branch-and-cut KW - optimization software Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10671 SN - 1438-0064 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 - 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 - 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 - Schlechte, Thomas T1 - Models for Railway Track Allocation N2 - This article is about the optimal track allocation problem (OPTRA) to find, in a given railway network, a conflict free set of train routes of maximum value. We study two types of integer programming formulations: a standard formulation that models block conflicts in terms of packing constraints, and a new extended formulation that is based on additional configuration' variables. We show that the packing constraints in the standard formulation stem from an interval graph, and that they can be separated in polynomial time. It follows that the LP relaxation of a strong version of this model, including all clique inequalities from block conflicts, can be solved in polynomial time. We prove that the extended formulation produces the same LP bound, and that it can also be computed with this model in polynomial time. Albeit the two formulations are in this sense equivalent, the extended formulation has advantages from a computational point of view, because it features a constant number of rows and is therefore amenable to standard column generation techniques. Results of an empirical model comparison on mesoscopic data for the Hannover-Fulda-Kassel region of the German long distance railway network are reported. T3 - ZIB-Report - 07-02 KW - Railway Slot Allocation KW - Train Timetabling KW - Combinatorial Optimization KW - Linear and Integer Programming Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9451 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 - 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 - 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 - Berthold, Timo T1 - Heuristics of the Branch-Cut-and-Price-Framework SCIP N2 - In this paper we give an overview of the heuristics which are integrated into the open source branch-cut-and-price-framework SCIP. We briefly describe the fundamental ideas of different categories of heuristics and present some computational results which demonstrate the impact of heuristics on the overall solving process of SCIP. T3 - ZIB-Report - 07-30 KW - Primalheuristiken KW - Gemischt-Ganzzahlige Programmierung KW - Branch and Cut KW - SCIP KW - mixed integer programming KW - primal heuristics KW - branch and cut KW - SCIP Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10280 SN - 1438-0064 ER - TY - THES A1 - Berthold, Timo T1 - Primal Heuristics for Mixed Integer Programs N2 - A lot of problems arising in Combinatorial Optimization and Operations Research can be formulated as Mixed Integer Programs (MIP). Although MIP-solving is an NP-hard optimization problem, many practically relevant instances can be solved in reasonable time. In modern MIP-solvers like the branch-cut-and-price-framework SCIP, primal heuristics play a major role in finding and improving feasible solutions at the early steps of the solution process. This helps to reduce the overall computational effort, guides the remaining search process, and proves the feasibility of the MIP model. Furthermore, a heuristic solution with a small gap to optimality often is sufficient in practice. We investigate 16 different heuristics, all of which are available in SCIP. Four of them arise from the literature of the last decade, nine are specific implementations of general heuristic ideas, three have been newly developed. We present an improved version of the feasibility pump heuristic by Fischetti et al., which in experiments produced solutions with only a third of the optimality gap compared to the original version. Furthermore, we introduce two new Large Neighborhood Search (LNS) heuristics. Crossover is an LNS improvement heuristic making use of similarities of diverse MIP solutions to generate new incumbent solutions. RENS is an LNS rounding heuristic which evaluates the space of all possible roundings of a fractional LP-solution. This heuristic makes it possible to determine whether a point can be rounded to an integer solution and which is the best possible rounding. We conclude with a computational comparison of all described heuristics. It points out that a single heuristic on its own has only a slight impact on the overall performance of SCIP, but the combination of all of them reduces the running time by a factor of two compared to a version without any heuristics. Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10293 ER - TY - THES A1 - Bley, Andreas T1 - Routing and Capacity Optimization for IP Networks N2 - This thesis is concerned with dimensioning and routing optimization problems for communication networks that employ a shortest path routing protocol such as OSPF, IS-IS, or RIP. These protocols are widely used in the Internet. With these routing protocols, all end-to-end data streams are routed along shortest paths with respect to a metric of link lengths. The network administrator can configure the routing only by modifying this metric. In this thesis we consider the unsplittable shortest path routing variant, where each communication demand must be sent unsplit through the network. This requires that all shortest paths are uniquely determined. The major difficulties in planning such networks are that the routing can be controlled only indirectly via the routing metric and that all routing paths depend on the same routing metric. This leads to rather complicated and subtle interdependencies among the paths that comprise a valid routing. In contrast to most other routing schemes, the paths for different communication demands cannot be configured independent of each other. Part I of the thesis is dedicated to the relation between path sets and routing metrics and to the combinatorial properties of those path sets that comprise a valid unsplittable shortest path routing. Besides reviewing known approaches to find a compatible metric for a given path set (or to prove that none exists) and discussing some properties of valid path sets, we show that the problem of finding a compatible metric with integer lengths as small as possible and the problem of finding a smallest possible conflict in the given path set are both NP-hard to approximate within a constant factor. In Part II of the thesis we discuss the relation between unsplittable shortest path routing and several other routing schemes and we analyze the computational complexity of three basic unsplittable shortest path routing problems. We show that the lowest congestion that can be obtained with unsplittable shortest path routing may significantly exceed that achievable with other routing paradigms and we prove several non-approximability results for unsplittable shortest path routing problems that are stronger than those for the corresponding unsplittable flow problems. In addition, we derive various polynomial time approximation algorithms for general and special cases of these problems. In Part III of the thesis we finally develop an integer linear programming approach to solve these and more realistic unsplittable shortest path routing problems to optimality. We present alternative formulations for these problems, discuss their strength and computational complexity, and show how to derive strong valid inequalities. Eventually, we describe our implementation of this solution approach and report on the numerical results obtained for real-world problems that came up in the planning the German National Research and Education Networks G-WiN and X-WiN and for several benchmark instances. N2 - Die Arbeit befasst sich mit der Kapazitäts- und Routenplanung für Kommunikationsnetze, die ein kürzeste-Wege Routingprotokoll verwenden. Diese Art von Protokollen ist im Internet weit verbreitet. Bei diesen Routingverfahren wird für jede Verbindung im Netz ein Längenwert festgelegt, diese Längen formen die sogenannte Routingmetrik. Die Routingwege der Kommunikationsbedarfe sind dann die jeweiligen kürzesten Wege bezüglich dieser Metrik. Bei der in der Arbeit untersuchten Variante dieser Routingprotokolle wird zusätzlich verlangt, dass es je Kommunikationsbedarf genau einen eindeutigen kürzesten Weg gibt. Die Schwierigkeit bei der Planung solcher Netze besteht darin, dass sich die Routingwege einerseits nur indirekt über die Routingmetrik beeinflussen lassen, andererseits aber alle Routingwege von der gleichen Metrik abhängen. Dadurch können die Wege verschiedener Kommunikationsanforderungen nicht wie bei anderen Routingverfahren unabhängig voneinander gewählt werden. Im erstem Teil der Arbeit werden der Zusammenhang zwischen gegebenen Wegesystemen und kompatiblen Routingmetriken sowie die Beziehungen der Wege eines zulässigen eindeutige-kürzeste-Wege-Routings untereinander untersucht. Dabei wird unter Anderem gezeigt, dass es NP-schwer ist, eine kompatible Metrik mit kleinstmöglichen Routinglängen zu einem gegebenen Wegesystem zu finden. Es wird auch bewiesen, dass das Finden eines kleinstmöglichen Konfliktes in einem gegebenen Wegesystem, zu dem keine kompatible Metrik existiert, NP-schwer ist. Im zweiten Teil der Arbeit wird die Approximierbarkeit von drei grundlegenden Netz- und Routenplanungsproblemen mit eindeutige-kürzeste-Wege-Routing untersucht. Für diese Probleme werden stärkere Nichtapproximierbarkeitsresultate als für die entsprechenden Einwege-Routing Probleme bewiesen und es werden verschiedene polynomiale Approximationsverfahren für allgemeine und Spezialfälle entworfen. Ausserdem wird die Beziehung zwischen eindeutige-kürzeste-Wege-Routing und anderen Routingverfahren diskutiert. Im dritten und letzten Teil der Arbeit wird ein (gemischt-) ganzzahliger Lösungsansatz für Planungsprobleme mit eindeutige-kürzeste-Wege-Routing vorgestellt. Für die im zweiten Teil diskutierten grundlegenden Netz- und Routenplanungsprobleme werden verschiedene (gemischt-) ganzzahlige lineare Modelle vorgestellt und es wird deren Lösbarkeit und die Stärke ihrer LP Relaxierungen untersucht. Es wird auch gezeigt, wie sich starke gültig Ungleichungen aus den in diesen Modellen enthalten Substrukturen ableiten lassen. Schlielich werden am Ende der Arbeit die Software-Implementierung dieses Lösungsverfahrens für eine praxisrelevante Verallgemeinerung der Planungsprobleme sowie die damit erzielten numerischen Ergebnisse vorgestellt und diskutiert. KW - kombinatorische Optimierung KW - gemischt-ganzzahlige Programmierung KW - kürzeste-Wege Routing KW - Approximationsalgorithmen KW - combinatorial optimization KW - mixed-integer programming KW - shortest path routing KW - approximation algorithms Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:83-opus-15530 SN - 978-3-86727-281-0 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 - THES A1 - Martin, Alexander T1 - Integer Programs with Block Structure N2 - In this thesis we study and solve integer programs with block structure, i.\,e., problems that after the removal of certain rows (or columns) of the constraint matrix decompose into independent subproblems. The matrices associated with each subproblem are called blocks and the rows (columns) to be removed linking constraints (columns). Integer programs with block structure come up in a natural way in many real-world applications. The methods that are widely used to tackle integer programs with block structure are decomposition methods. The idea is to decouple the linking constraints (variables) from the problem and treat them at a superordinate level, often called master problem. The resulting residual subordinate problem then decomposes into independent subproblems that often can be solved more efficiently. Decomposition methods now work alternately on the master and subordinate problem and iteratively exchange information to solve the original problem to optimality. In Part I we follow a different approach. We treat the integer programming problem as a whole and keep the linking constraints in the formulation. We consider the associated polyhedra and investigate the polyhedral consequences of the involved linking constraints. The variety and complexity of the new inequalities that come into play is illustrated on three different types of real-world problems. The applications arise in the design of electronic circuits, in telecommunication and production planning. We develop a branch-and-cut algorithm for each of these problems, and our computational results show the benefits and limits of the polyhedral approach to solve these real-world models with block structure. Part II of the thesis deals with general mixed integer programming problems, that is integer programs with no apparent structure in the constraint matrix. We will discuss in Chapter 5 the main ingredients of an LP based branch-and-bound algorithm for the solution of general integer programs. Chapter 6 then asks the question whether general integer programs decompose into certain block structures and investigate whether it is possible to recognize such a structure. The remaining two chapters exploit information about the block structure of an integer program. In Chapter 7 we parallelize parts of the dual simplex algorithm, the method that is commonly used for the solution of the underlying linear programs within a branch-and-cut algorithm. In Chapter 8 we try to detect small blocks in the constraint matrix and to derive new cutting planes that strengthen the integer programming formulation. These inequalities may be associated with the intersection of several knapsack problems. We will see that they significantly improve the quality of the general integer programming solver introduced in Chapter 5. T3 - ZIB-Report - SC-99-03 KW - integer programming KW - cutting planes KW - block structure Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3911 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 - THES A1 - Helmberg, Christoph T1 - Semidefinite Programming for Combinatorial Optimization N2 - This book offers a self-contained introduction to the field of semidefinite programming, its applications in combinatorial optimization, and its computational methods. We equip the reader with the basic results from linear algebra on positive semidefinite matrices and the cone spanned by them. Starting from linear programming, we introduce semidefinite programs and discuss the associated duality theory. We then turn to semidefinite relaxations of combinatorial optimization and illustrate their interrelation. In the second half we deal with computational methods for solving semidefinite programs. First, the interior point approach, its iteration complexity, and implementational issues are discussed. Next, we explain in great detail the spectral bundle method, which is particularly suited for large scale semidefinite programming. One of the most successful techniques in integer linear programming is the cutting plane approach which improves an initial relaxation by adding violated inequalities. We explore possibilities to combine the two solution methods with the cutting plane approach in order to strengthen semidefinite relaxations of combinatorial optimization problems. T3 - ZIB-Report - 00-34 KW - positive semidefinite matrices KW - semidefinite cone KW - semidefinite Programming KW - semidefinite duality KW - combinatorial optimization KW - max-cut KW - quadrati Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6022 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 - 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 - Borndörfer, Ralf A1 - Schlechte, Thomas T1 - Solving Railway Track Allocation Problems N2 - The \emph{optimal track allocation problem} (\textsc{OPTRA}), also known as the train routing problem or the train timetabling problem, is to find, in a given railway network, a conflict-free set of train routes of maximum value. We propose a novel integer programming formulation for this problem that is based on additional configuration' variables. Its LP-relaxation can be solved in polynomial time. These results are the theoretical basis for a column generation algorithm to solve large-scale track allocation problems. Computational results for the Hanover-Kassel-Fulda area of the German long distance railway network involving up to 570 trains are reported. T3 - ZIB-Report - 07-20 KW - railway track allocation KW - timetabling KW - column generation Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9631 SN - 1438-0064 ER - TY - GEN A1 - Kaibel, Volker A1 - Stephan, Rüdiger T1 - On cardinality constrained cycle and path polytopes N2 - We consider polytopes associated with cardinality constrained path and cycle problems defined on a directed or undirected graph. We present integer characterizations of these polytopes by facet defining linear inequalities for which the separation problem can be solved in polynomial time. Moreover, we give further facet defining inequalities, in particular those that are specific to odd/even paths and cycles. T3 - ZIB-Report - 07-25 KW - path KW - cycles KW - integer characterizations KW - facets Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10093 SN - 1438-0064 ER - TY - GEN A1 - Bley, Andreas T1 - Routing and Capacity Optimization for IP networks N2 - This article describes the main concepts and techniques that have been developed during the last year at ZIB to solve dimensioning and routing optimization problems for IP networks. We discuss the problem of deciding if a given path set corresponds to an unsplittable shortest path routing, the fundamental properties of such path sets, and the computational complexity of some basic network planning problems for this routing type. Then we describe an integer-linear programming approach to solve such problems in practice. This approach has been used successfully in the planning of the German national education and research network for several years. T3 - ZIB-Report - 07-33 KW - kombinatorische Optimierung KW - gemischt-ganzzahlige Programmierung KW - kürzeste-Wege Routing KW - Approximationsalgorithmen KW - combinatorial optimization KW - mixed-integer programming KW - shortest path routing KW - approximation algorithms Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10323 SN - 1438-0064 ER - TY - THES A1 - Weider, Steffen T1 - Integration of Vehicle and Duty Scheduling in Public Transport N2 - This thesis describes the algorithm IS-OPT that integrates scheduling of vehicles and duties in public bus transit. IS-OPT is the first algorithm which solves integrated vehicle and duty scheduling problems arising in medium sized carriers such that its solutions can be used in daily operations without further adaptions. This thesis is structured as follows: The first chapter highlights mathematical models of the planning process of public transit companies and examines their potential for integrating them with other planning steps. It also introduces descriptions of the vehicle and the duty scheduling problem. Chapter 2 motivates why it can be useful to integrate vehicle and duty scheduling, explains approaches of the literature, and gives an outline of our algorithm IS-OPT. The following chapters go into the details of the most important techniques and methods of IS-OPT: In Chapter 3 we describe how we use Lagrangean relaxation in a column generation framework. Next, in Chapter 4, we describe a variant of the proximal bundle method (PBM) that is used to approximate linear programs occurring in the solution process. We introduce here a new variant of the PBM which is able to utilize inexact function evaluation and the use of epsilon-subgradients. We also show the convergence of this method under certain assumptions. Chapter 5 treats the generation of duties for the duty scheduling problem. This problem is modeled as a resourceconstraint- shortest-path-problem with non-linear side constraints and nearly linear objective function. It is solved in a two-stage approach. At first we calculate lower bounds on the reduced costs of duties using certain nodes by a new inexact label-setting algorithm. Then we use these bounds to speed up a depth-first-search algorithm that finds feasible duties. In Chapter 6 we present the primal heuristic of IS-OPT that solves the integrated problem to integrality. We introduce a new branch-and-bound based heuristic which we call rapid branching. Rapid branching uses the proximal bundle method to compute lower bounds, it introduces a heuristic node selection scheme, and it utilizes a new branching rule that fixes sets of many variables at once. The common approach to solve the problems occurring in IS-OPT is to trade inexactness of the solutions for speed of the algorithms. This enables, as we show in Chapter 7, to solve large real world integrated problems by IS-OPT. The scheduled produced by IS-OPT save up to 5% of the vehicle and duty cost of existing schedules of regional and urban public transport companies. N2 - Diese Arbeit beschreibt den Algorithmus IS-OPT, welcher der erste Algorithmus ist, der integrierte Umlauf- und Dienstplanungsprobleme für mittelgroße Verkehrsunternehmen löst und dabei alle betrieblichen Einzelheiten berücksichtigt. Seine Lösungen können daher direkt in den täglichen Betrieb übernommen werden. Im ersten Kapitel werden mathematische Modelle für verschiedenen Probleme aus dem Planungsprozess von Nahverkehrsunternehmen beschrieben. Es werden Ansätze zur Integration der einzelnen Probleme untersucht. In diesem Kapitel werden außerdem das Umlauf- und das Dienstplanungsproblem eingeführt. Kapitel 2 motiviert, warum Integration von Umlauf- und Dienstplanung hilfreich ist oder in welchen Fällen sie sogar unabdingbar ist; es gibt einen Überblick über die vorhanden Literatur zur integrierten Umlauf- und Dienstplanung und umreißt unseren Algorithmus IS-OPT. Die weiteren Kapitel behandeln die in IS-OPT verwendeten Methoden: In Kapitel 3 beschreiben wir, wie Spaltenerzeugung für lineare Programme mit Lagrange-Relaxierung und Subgradienten-Verfahren kombiniert werden kann. In Kapitel 4 wird unsere Variante der proximalen Bündelmethode beschrieben. Diese wird benutzt um näherungsweise primale und duale Lösungen von lineare Programmen zu berechnen. Unsere Variante ist angepasst, um auch mit ungenauer Funktionsauswertung und Epsilon-Subgradienten arbeiten zu können. Wir zeigen die Konvergenz dieser Variante unter bestimmten Annahmen. Kapitel 5 behandelt das Erzeugen von Diensten für das Dienstplanungsproblem. Dieses Problem ist als ein Kürzeste-Wege-Problem mit nichtlinearen Nebenbedingungen und fast linearer Zielfunktion modelliert. Wir lösen es, indem zuerst Schranken für die reduzierten Kosten von Diensten, die bestimmte Knoten benutzen, berechnet werden. Diese Schranken werden benutzt, um in einem Tiefensuchalgorithmus den Suchbaum klein zu halten. Im Kapitel 6 präsentieren wir die neu entwickelte Heuristik "Rapid Branching", die eine ganzzahlige Lösung des integrierten Problems berechnet. Rapid Branching kann als eine spezielle Branch-and-Bound-Heuristik gesehen werden, welche die Bündelmethode benutzt. In den Knoten des Suchbaums können mehrere Variablen auf einmal fixiert werden, die mit Hilfe einer Perturbationsheuristik ausgewählt werden. In Kapitel 7 schließlich zeigen wir, daß wir mit IS-OPT auch große Probleminstanzen aus der Praxis lösen können und dabei bis zu 5% der Fahrzeug- und Dienstkosten sparen können. KW - Öffentlicher Personennahverkehr KW - lineare Optimierung KW - Umlaufplanung KW - Dienstplanung KW - Integration KW - Public Transport KW - linear Optimization KW - vehicle scheduling KW - duty scheduling KW - integration Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:83-opus-16240 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 - Poss, Michael A1 - Raack, Christian T1 - Affine recourse for the robust network design problem: between static and dynamic routing N2 - Affinely-Adjustable Robust Counterparts provide tractable alternatives to (two-stage) robust programs with arbitrary recourse. We apply them to robust network design with polyhedral demand uncertainty, introducing the affine routing principle. We compare the affine routing to the well-studied static and dynamic routing schemes for robust network design. All three schemes are embedded into the general framework of two-stage network design with recourse. It is shown that affine routing can be seen as a generalization of the widely used static routing still being tractable and providing cheaper solutions. We investigate properties on the demand polytope under which affine routings reduce to static routings and also develop conditions on the uncertainty set leading to dynamic routings being affine. We show however that affine routings suffer from the drawback that (even totally) dominated demand vectors are not necessarily supported by affine solutions. Uncertainty sets have to be designed accordingly. Finally, we present computational results on networks from SNDlib. We conclude that for these instances the optimal solutions based on affine routings tend to be as cheap as optimal network designs for dynamic routings. In this respect the affine routing principle can be used to approximate the cost for two-stage solutions with free recourse which are hard to compute. T3 - ZIB-Report - 11-03 KW - robust optimization KW - network design KW - recourse KW - affine adjustable robust counterparts KW - affine routing Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-12122 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Neumann, Marika T1 - Linienoptimierung - reif für die Praxis? N2 - Wir stellen in dieser Arbeit ein mathematisches Optimierungsmodell zur Bestimmung eines optimalen Linienplans vor, das sowohl die Fahrzeiten und die Anzahl der Umstiege berücksichtigt als auch die Kosten des Liniennetzes. Dieses Modell deckt wichtige praktische Anforderungen ab, die in einem gemeinsamen Projekt mit den Verkehrsbetrieben in Potsdam (ViP) formuliert wurden. In diesem Projekt wurde der Linienplan 2010 für Potsdam entwickelt. Unsere Berechnungen zeigen, dass die mathematische Optimierung in nichts einer "Handplanung" des Liniennetzes nachsteht. Im Gegenteil, mit Hilfe des Optimierungsprogramms ist es möglich, durch Veränderung der Parameter mehrere verschiedene Szenarien zu berechnen, miteinander zu vergleichen und Aussagen über minimale Kosten und Fahrzeiten zu machen. T3 - ZIB-Report - 10-20 KW - Linienplanung KW - Optimierung Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11832 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Schlechte, Thomas A1 - Weider, Steffen T1 - Railway Track Allocation by Rapid Branching N2 - The track allocation problem, also known as train routing problem or train timetabling problem, is to find a conflict-free set of train routes of maximum value in a railway network. Although it can be modeled as a standard path packing problem, instances of sizes relevant for real-world railway applications could not be solved up to now. We propose a rapid branching column generation approach that integrates the solution of the LP relaxation of a path coupling formulation of the problem with a special rounding heuristic. The approach is based on and exploits special properties of the bundle method for the approximate solution of convex piecewise linear functions. Computational results for difficult instances of the benchmark library TTPLIB are reported. T3 - ZIB-Report - 10-22 KW - Trassenallokationsproblem KW - Ganzzahlige Programmierung KW - Rapid-Branching-Heuristik KW - Proximale Bündelmethode KW - Optimization KW - Railway Track Allocation KW - Bundle Method KW - Rapid Branching Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11864 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Erol, Berkan A1 - Graffagnino, Thomas A1 - Schlechte, Thomas A1 - Swarat, Elmar T1 - Optimizing the Simplon Railway Corridor N2 - This paper presents a case study on a railway timetable optimization for the very dense Simplon corridor, a major railway connection in the Alps between Switzerland and Italy. Starting from a detailed microscopic network as it is used in railway simulation, the data is transformed by an automatic procedure to a less detailed macroscopic network, that is sufficient for the purpose of capacity planning and amenable to state-of-the-art integer programming optimization methods. In this way, the macroscopic railway network is saturated with trains. Finally, the corresponding timetable is re-transformed to the microscopic level in such a way that it can be operated without any conflicts among the slots. Using this integer programming based micro-macro aggregation-disaggregation approach, it becomes for the first time possible to generate a profit maximal and conflict free timetable for the complete Simplon corridor over an entire day by a simultaneous optimization of all trains requests. This also allows to to undertake a sensitivity analysis of various problem parameters. T3 - ZIB-Report - 10-24 KW - Trassenallokation KW - Netzwerkaggregation KW - Fallstudie KW - Simplon Korridor KW - railway track allocation KW - network aggregation KW - case study KW - Simplon corridor Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11892 SN - 1438-0064 ER - TY - THES A1 - Heismann, Olga T1 - The Hypergraph Assignment Problem N2 - This thesis deals with the hypergraph assignment problem (HAP), a set partitioning problem in a special type of hypergraph. The HAP generalizes the assignment problem from bipartite graphs to what we call bipartite hypergraphs, and is motivated by applications in railway vehicle rotation planning. The main contributions of this thesis concern complexity, polyhedral results, analyses of random instances, and primal methods for the HAP. We prove that the HAP is NP-hard and APX-hard even for small hyperedge sizes and hypergraphs with a special partitioned structure. We also study the complexity of the set packing and covering relaxations of the HAP, and present for certain cases polynomial exact or approximation algorithms. A complete linear description is known for the assignment problem. We therefore also study the HAP polytope. There, we have a huge number of facet-defining inequalities already for a very small problem size. We describe a method for dividing the inequalities into equivalence classes without resorting to a normal form. Within each class, facets are related by certain symmetries and it is sufficient to list one representative of each class to give a complete picture of the structural properties of the polytope. We propose the algorithm "HUHFA" for the classification that is applicable not only to the HAP but combinatorial optimization problems involving symmetries in general. In the largest possible HAP instance for which we could calculate the complete linear description, we have 14049 facets, which can be divided into 30 symmetry classes. We can combinatorially interpret 16 of these classes. This is possible by employing cliques to generalize the odd set inequalities for the matching problem. The resulting inequalities are valid for the polytope associated with the set packing problem in arbitrary hypergraphs and have a clear combinatorial meaning. An analysis of random instances provides a better insight into the structure of hyperassignments. Previous work has extensively analyzed random instances for the assignment problem theoretically and practically. As a generalization of these results for the HAP, we prove bounds on the expected value of a minimum cost hyperassignment that uses half of the maximum possible number of hyperedges that are not edges. In a certain complete partitioned hypergraph G2,2n with i. i. d. exponential random variables with mean 1 as hyperedge costs it lies between 0.3718 and 1.8310 if the vertex number tends to infinity. Finally, we develop an exact combinatorial solution algorithm for the HAP that combines three methods: A very large-scale neighborhood search, the composite columns method for the set partitioning problem, and the network simplex algorithm. N2 - Diese Arbeit beschäftigt sich mit dem Hypergraph Assignment Problem (Abkürzung "HAP", dt.: Zuordnungsproblem auf Hypergraphen), einem Mengenzerlegungsproblem auf einem speziellen Typ von Hypergraphen. Das HAP verallgemeinert das Zuordnungsproblem von bipartiten Graphen auf eine Struktur, die wir bipartite Hypergraphen nennen, und ist durch eine Anwendung in der Umlaufplanung im Schienenverkehr motiviert. Die Hauptresultate betreffen die Komplexität, polyedrische Ergebnisse, die Analyse von Zufallsinstanzen sowie primale Methoden für das HAP. Wir beweisen, dass das HAP NP-schwer und APX-schwer ist, sogar wenn wir uns auf kleine Hyperkantengrößen und Hypergraphen mit einer speziellen, partitionierten Struktur beschränken. Darüber hinaus untersuchen wir die Komplexität der Mengenpackungs- sowie Mengenüberdeckungsrelaxierung und geben für bestimmte Fälle Approximations- und exakte Algorithmen mit einer polynomiellen Laufzeit an. Für das Polytop des Zuordnungsproblems ist eine vollständige lineare Beschreibung bekannt. Wir untersuchen daher auch das HAP-Polytop. Dafür ist die Anzahl der Facettenungleichungen schon für sehr kleine Problemgrößen sehr groß. Wir beschreiben eine Methode zur Aufteilung der Ungleichungen in Äquivalenzklassen, die ohne die Verwendung von Normalformen auskommt. Die Facetten in jeder Klasse können durch Symmetrien ineinander überführt werden. Es genügt, einen Repräsentanten aus jeder Klasse anzugeben, um ein vollständiges Bild der Polytopstruktur zu erhalten. Wir beschreiben den Algorithmus "HUHFA", der diese Klassifikation nicht nur für das HAP, sondern für beliebige kombinatorische Optimierungsprobleme, die Symmetrien enthalten, durchführt. Die größtmögliche HAP-Instanz, für die wir die vollständige lineare Beschreibung berechnen konnten, hat 14049 Facetten, die in 30 Symmetrieklassen aufgeteilt werden können. Wir können 16 dieser Klassen kombinatorisch interpretieren. Dafür verallgemeinern wir Odd-Set-Ungleichungen für das Matchingproblem unter Verwendung von Cliquen. Die Ungleichungen, die wir erhalten, sind gültig für Mengenpackungsprobleme in beliebigen Hypergraphen und haben eine klare kombinatorische Bedeutung. Die Analyse von Zufallsinstanzen erlaubt einen besseren Einblick in die Struktur von Hyperzuordnungen. Eine solche ausführliche Analyse wurde in der Literatur theoretisch und praktisch bereits für das Zuordnungsproblem durchgeführt. Als eine Verallgemeinerung dieser Ergebnisse für das HAP beweisen wir Schranken für den Erwartungswert einer Hyperzuordnung mit minimalen Kosten, die genau die Hälfte der maximal möglichen Anzahl an Hyperkanten, die keine Kanten sind, benutzt. In einem sog. vollständigen partitionierten Hypergraphen G2,2n mit Hyperkantenkosten, die durch unabhängig identisch exponentiell verteilte Zufallsvariablen mit Erwartungswert 1 bestimmt sind, liegt dieser Wert zwischen 0.3718 und 1.8310, wenn die Knotenanzahl gegen unendlich strebt. Schließlich entwickeln wir eine exakte kombinatorische Lösungsmethode für das HAP, die drei Ansätze kombiniert: Eine Nachbarschaftssuche mit Nachbarschaften exponentieller Größe, die Composite-Columns-Methode für das Mengenzerlegungsproblem sowie den Netzwerksimplexalgorithmus. KW - assignment KW - bipartite KW - hyperassignment KW - hypergraph KW - set partitioning Y1 - 2014 UR - http://opus4.kobv.de/opus4-tuberlin/frontdoor/index/index/docId/5321 ER - TY - GEN A1 - Heismann, Olga A1 - Borndörfer, Ralf T1 - A Generalization of Odd Set Inequalities for the Set Packing Problem N2 - The set packing problem, sometimes also called the stable set problem, is a well-known NP-hard problem in combinatorial optimization with a wide range of applications and an interesting polyhedral structure, that has been the subject of intensive study. We contribute to this field by showing how, employing cliques, odd set inequalities for the matching problem can be generalized to valid inequalities for the set packing polytope with a clear combinatorial meaning. T3 - ZIB-Report - 14-28 KW - set packing KW - matching KW - hypergraph KW - inequality KW - facet Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-51010 SN - 1438-0064 ER - TY - GEN A1 - Hoppmann, Heide T1 - An Extended Formulation for the Line Planning Problem N2 - In this paper we present a novel extended formulation for the line planning problem that is based on what we call “configurations” of lines and frequencies. Configurations account for all possible options to provide a required transportation capacity on an infrastructure edge. The proposed configuration model is strong in the sense that it implies several facet-defining inequalities for the standard model: set cover, symmetric band, MIR, and multicover inequalities. These theoretical findings can be confirmed in computational results. Further, we show how this concept can be generalized to define configurations for subsets of edges; the generalized model implies additional inequalities from the line planning literature. T3 - ZIB-Report - 16-08 KW - Line Planning, Extended Formulation, Public Transport Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-57467 SN - 1438-0064 ER - TY - THES A1 - Karbstein, Marika T1 - Line Planning and Connectivity N2 - This thesis introduces the Steiner connectivity problem. It is a generalization of the well known Steiner tree problem. Given a graph G = (V, E) and a subset T ⊆ V of the nodes, the Steiner tree problem consists in finding a cost minimal set of edges connecting all nodes in T . The Steiner connectivity problem chooses, instead of edges, from a given set of paths a subset to connect all nodes in T . We show in the first part of this thesis that main results about complexity, approximation, integer programming formulations, and polyhedra can be generalized from the Steiner tree problem to the Steiner connectivity problem. An example for a straightforward generalization are the Steiner partition inequalities, a fundamental class of facet defining inequalities for the Steiner tree problem. They can be defined for the Steiner connectivity problem in an analogous way as for the Steiner tree problem. An example for a generalization that needs more effort is the definition of a directed cut formulation and the proof that this dominates the canonical undirected cut formulation enriched by all Steiner partition inequalities. For the Steiner connectivity problem this directed cut formulation leads to extended formulations, a concept that is not necessary for the Steiner tree problem. There are also major differences between both problems. For instance, the case T = V for the Steiner connectivity problem is equivalent to a set covering problem and, hence, not a polynomial solvable case as in the Steiner tree problem. The Steiner connectivity problem is not only an interesting generalization of the Steiner tree problem but also the underlying connectivity problem in line planning with inte- grated passenger routing. The integrated line planning and passenger routing problem is an important planning problem in service design of public transport and the topic of the second part. Given is the infrastructure network of a public transport system where the edges correspond to streets and tracks and the nodes correspond to stations/stops of lines. The task is to find paths in the infrastructure network for lines and passengers such that the capacities of the lines suffice to transport all passengers. Existing models in the literature that integrate a passenger routing in line planning either treat transfers in a rudimentary way and, hence, neglect an important aspect for the choice of the pas- senger routes, or they treat transfers in a too comprehensive way and cannot be solved for large scale real world problems. We propose a new model that focuses on direct connections. The attractiveness of transfer free connections is increased by introducing a transfer penalty for each non-direct connection. In this way, a passenger routing is computed that favors direct connections. For the computation of this model we also implemented algorithms influenced by the results for the Steiner connectivity problem. We can compute with our model good solutions that minimize a weighted sum of line operating costs and passengers travel times. These solutions improve the solutions of an existing approach, that does not consider direct connections, by up to 17%. In contrast to a comprehensive approach, that considers every transfer and for which we could not even solve the root LP within 10 hours for large instances, the solutions of the new model, computed in the same time, are close to optimality (<1%) or even optimal for real world instances. In a project with the Verkehr in Potsdam GmbH to compute the line plan for 2010 we showed that our approach is applicable in practice and can be used to solve real world problems. Y1 - 2013 UR - www.zib.de/karbstein/scplpp.pdf SN - 978-3-8439-1062-0 ER - TY - GEN A1 - Heismann, Olga A1 - Hildenbrandt, Achim A1 - Silvestri, Francesco A1 - Reinelt, Gerhard A1 - Borndörfer, Ralf T1 - HUHFA: A Framework for Facet Classification N2 - Usually complete linear descriptions of polytopes consist of an enormous number of facet-defining inequalities already for very small problem sizes. In this paper, we describe a method for dividing the inequalities into equivalence classes without resorting to a normal form. Within each class, facets are related by certain symmetries and it is sufficient to list one representative of each class to give a complete picture of the structural properties of a polytope. We propose an algorithm for the classification and illustrate its efficiency on a broad range of combinatorial optimization problems including the Traveling Salesman and the Linear Ordering Problem. T3 - ZIB-Report - 13-45 KW - facet classification KW - symmetry KW - polyhedral structure Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-42230 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Hoppmann, Heide A1 - Karbstein, Marika T1 - Umsteigen ohne Warten N2 - Wir stellen einen mathematischen Optimierungsansatz zur Berechnung von periodischen Taktfahrplänen vor, bei dem die Umsteigezeiten unter Berücksichtigung des Passagierverhaltens minimiert werden. Wir untersuchen damit den Einfluss wichtiger Systemparameter und Verhaltensmuster auf die Beförderungsqualität. T3 - ZIB-Report - 17-05 Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-62006 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Hoppmann, Heide A1 - Karbstein, Marika T1 - A Configuration Model for the Line Planning Problem N2 - We propose a novel extended formulation for the line planning problem in public transport. It is based on a new concept of frequency configurations that account for all possible options to provide a required transportation capacity on an infrastructure edge. We show that this model yields a strong LP relaxation. It implies, in particular, general classes of facet defining inequalities for the standard model. T3 - ZIB-Report - 13-40 KW - combinatorial optimization KW - polyhedral combinatorics KW - line planning Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-41903 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Hoppmann, Heide A1 - Karbstein, Marika T1 - Passenger Routing for Periodic Timetable Optimization N2 - The task of periodic timetabling is to schedule the trips in a public transport system by determining arrival and departure times at every station such that travel and transfer times are minimized. To date, the optimization literature generally assumes that passengers do not respond to changes in the timetable, i.e., the passenger routes are fixed. This is unrealistic and ignores potentially valuable degrees of freedom. We investigate in this paper periodic timetabling models with integrated passenger routing. We show that different routing models have a huge influence on the quality of the entire system: Whatever metric is applied, the performance ratios of timetables w.r.t. to different routing models can be arbitrarily large. Computations on a real-world instance for the city of Wuppertal substantiate the theoretical findings. These results indicate the existence of untapped optimization potentials that can be used to improve the efficiency of public transport systems. T3 - ZIB-Report - 15-55 KW - Passenger routing KW - Periodic timetabling KW - Public transport Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-56739 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Karbstein, Marika T1 - Metric Inequalities for Routings on Direct Connections N2 - We consider multi-commodity flow problems in which capacities are installed on paths. In this setting, it is often important to distinguish between flows on direct connection routes, using single paths, and flows that include path switching. We show that the concept of metric inequalities to characterize capacities that support a multi-commodity flow can be generalized to deal with direct connections. T3 - ZIB-Report - 14-04 KW - metric inequalities KW - direct connection KW - multi-commodity flow problem Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-44219 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Karbstein, Marika T1 - Metric Inequalities for Routings on Direct Connections with Application in Line Planning N2 - We consider multi-commodity flow problems in which capacities are installed on paths. In this setting, it is often important to distinguish between flows on direct connection routes, using single paths, and flows that include path switching. We derive a feasibility condition for path capacities supporting such direct connection flows similar to the feasibility condition for arc capacities in ordinary multi-commodity flows. The concept allows to solve large-scale real-world line planning problems in public transport including a novel passenger routing model that favors direct connections over connections with transfers. T3 - ZIB-Report - 15-07 KW - combinatorial optimization KW - integer programming KW - line planning KW - transfers KW - direct connection KW - metric inequalities Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-53507 SN - 1438-0064 ER - TY - GEN A1 - Karbstein, Marika T1 - Integrated Line Planning and Passenger Routing: Connectivity and Transfers N2 - The integrated line planning and passenger routing problem is an important planning problem in service design of public transport. A major challenge is the treatment of transfers. A main property of a line system is its connectivity. In this paper we show that analysing the connecvitiy aspect of a line plan gives a new idea to handle the transfer aspect of the line planning problem. T3 - ZIB-Report - 14-42 KW - line planning, transfers,Steiner connectivity problem Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-52986 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 - Borndörfer, Ralf A1 - Hoppmann, Heide A1 - Karbstein, Marika T1 - Timetabling and Passenger Routing in Public Transport T2 - Appeard in: Proceedings of Conference on Advanced Systems in Public Transport 2015 (CASPT2015) N2 - The task of timetabling is to schedule the trips in a public transport system by determining periodic arrival and departure times at every station. The goal is to provide a service that is both attractive for passengers and can be operated economically. To date, timetable optimization is generally done with respect to fixed passenger routes, i.e., it is assumed that passengers do not respond to changes in the timetable. This is unrealistic and ignores potentially valuable degrees of freedom. We investigate in this paper periodic timetabling models with integrated passenger routing. We propose several models that differ in the allowed passenger paths and the objectives. We compare these models theoretically and report on computations on real-world instances for the city of Wuppertal. T3 - ZIB-Report - 15-31 KW - Passenger routing KW - Periodic timetabling KW - Public transport Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-55305 SN - 1438-0064 ER - TY - GEN A1 - Gamrath, Gerald A1 - Koch, Thorsten A1 - Rehfeldt, Daniel A1 - Shinano, Yuji T1 - SCIP-Jack - A massively parallel STP solver N2 - In this article we describe the impact from embedding a 15 year old model for solving the Steiner tree problem in graphs in a state-of-the-art MIP-Framework, making the result run in a massively parallel environment and extending the model to solve as many variants as possible. We end up with a high-perfomance solver that is capable of solving previously unsolved instances and, in contrast to its predecessor, is freely available for academic research. T3 - ZIB-Report - 14-35 KW - Steiner Tree Problem KW - Mixed-Integer Programming KW - Branch-and-Cut KW - DIMACS Challenge Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-52293 SN - 1438-0064 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 - 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 - 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 - 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 - 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 - GEN A1 - Borndörfer, Ralf A1 - Danecker, Fabian A1 - Weiser, Martin T1 - A Discrete-Continuous Algorithm for Free Flight Planning N2 - We propose a hybrid discrete-continuous algorithm for flight planning in free flight airspaces. In a first step, our DisCOptER method discrete-continuous optimization for enhanced resolution) computes a globally optimal approximate flight path on a discretization of the problem using the A* method. This route initializes a Newton method that converges rapidly to the smooth optimum in a second step. The correctness, accuracy, and complexity of the method are goverened by the choice of the crossover point that determines the coarseness of the discretization. We analyze the optimal choice of the crossover point and demonstrate the asymtotic superority of DisCOptER over a purely discrete approach. T3 - ZIB-Report - 20-33 KW - shortest path KW - flight planning KW - free flight KW - discrete-continuous algorithm KW - optimal control KW - discrete optimization Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-81343 SN - 1438-0064 ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Danecker, Fabian A1 - Weiser, Martin T1 - A Discrete-Continuous Algorithm for Free Flight Planning JF - Algorithms N2 - We propose a hybrid discrete-continuous algorithm for flight planning in free flight airspaces. In a first step, our DisCOptER method discrete-continuous optimization for enhanced resolution) computes a globally optimal approximate flight path on a discretization of the problem using the A* method. This route initializes a Newton method that converges rapidly to the smooth optimum in a second step. The correctness, accuracy, and complexity of the method are goverened by the choice of the crossover point that determines the coarseness of the discretization. We analyze the optimal choice of the crossover point and demonstrate the asymtotic superority of DisCOptER over a purely discrete approach. KW - shortest path KW - flight planning KW - free flight KW - discrete-continuous algorithm KW - optimal control KW - discrete optimization Y1 - 2020 U6 - https://doi.org/10.3390/a14010004 SN - 1438-0064 VL - 14 IS - 1 SP - 4 PB - MDPI ER - TY - GEN A1 - Bortoletto, Enrico A1 - Lindner, Niels A1 - Masing, Berenike T1 - The tropical and zonotopal geometry of periodic timetables N2 - The Periodic Event Scheduling Problem (PESP) is the standard mathematical tool for optimizing periodic timetabling problems in public transport. A solution to PESP consists of three parts: a periodic timetable, a periodic tension, and integer periodic offset values. While the space of periodic tension has received much attention in the past, we explore geometric properties of the other two components, establishing novel connections between periodic timetabling and discrete geometry. Firstly, we study the space of feasible periodic timetables, and decompose it into polytropes, i.e., polytopes that are convex both classically and in the sense of tropical geometry. We then study this decomposition and use it to outline a new heuristic for PESP, based on the tropical neighbourhood of the polytropes. Secondly, we recognize that the space of fractional cycle offsets is in fact a zonotope. We relate its zonotopal tilings back to the hyperrectangle of fractional periodic tensions and to the tropical neighbourhood of the periodic timetable space. To conclude we also use this new understanding to give tight lower bounds on the minimum width of an integral cycle basis. T3 - ZIB-Report - 22-09 KW - periodic event scheduling KW - tropical geometry KW - zonotopal tilings Y1 - 2022 SN - 1438-0064 ER -