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 -