TY - GEN A1 - Bley, Andreas A1 - Koch, Thorsten A1 - Wessäly, Roland T1 - Large-scale hierarchical networks: How to compute an optimal architecture? N2 - In this article, we present a mathematical model and an algorithm to support one of the central strategic planning decisions of network operators: How to organize a large number of locations into a hierarchical network? We propose a solution approach that is based on mixed-integer programming and Lagrangian relaxation techniques. As major advantage, our approach provides not only solutions but also worst-case quality guarantees. Real-world scenarios with more than 750 locations have been solved within 30 minutes to less than 1\% off optimality. T3 - ZIB-Report - 04-04 KW - Hierarchical Networks KW - Mixed-Integer Programming KW - Lagrangian Relaxation Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7799 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 - Bley, Andreas A1 - Pattloch, Marcus T1 - Modellierung und Optimierung der X-WiN Plattform N2 - In diesem Artikel werden die Optimierungsmodelle und -verfahren beschrieben, die bei der Planung des Kernnetzes und der Zugangsinfrastruktur des X-WiN verwendet wurden. T3 - ZIB-Report - 05-21 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8551 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 - Bley, Andreas A1 - Grötschel, Martin A1 - Wessäly, Roland T1 - Design of Broadband Virtual Private Networks: Model and Heuristics for the B–WiN N2 - We investigate the problem of designing survivable broadband virtual private networks that employ the Open Shortest Path First (OSPF) routing protocol to route the packages. The capacities available for the links of the network are a minimal capacity plus multiples of a unit capacity. Given the directed communication demands between all pairs of nodes, we wish to select the capacities in a such way, that even in case of a single node or a single link failure a specified percentage of each demand can be satisfied and the costs for these capacities are minimal. We present a mixed--integer linear programming formulation of this problem and several heuristics for its solution. Furthermore, we report on computational results with real-world data. T3 - ZIB-Report - SC-98-13 KW - Telecommunication Network Design KW - Survivable Networks KW - Network Capacity Planning KW - OSPF Routing KW - Shortest Path Routing KW - Heuristics Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3565 ER - TY - GEN A1 - Bley, Andreas A1 - Gleixner, Ambros A1 - Koch, Thorsten A1 - Vigerske, Stefan T1 - Comparing MIQCP solvers to a specialised algorithm for mine production scheduling N2 - In this paper we investigate the performance of several out-of-the box solvers for mixed-integer quadratically constrained programmes (MIQCPs) on an open pit mine production scheduling problem with mixing constraints. We compare the solvers BARON, Couenne, SBB, and SCIP to a problem-specific algorithm on two different MIQCP formulations. The computational results presented show that general-purpose solvers with no particular knowledge of problem structure are able to nearly match the performance of a hand-crafted algorithm. T3 - ZIB-Report - 09-32 Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11507 SN - 1438-0064 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 - Bley, Andreas A1 - Koster, Arie M.C.A. A1 - Kröller, Alexander A1 - Wessäly, Roland A1 - Zymolka, Adrian T1 - Kosten- und Qualitätsoptimierung in Kommunikationsnetzen N2 - Der scharfe Wettbewerb innerhalb der Telekommunikationsbranche zwingt die Netzbetreiber dazu, ihre Investitionen genau zu planen und immer wieder Einsparungsmanahmen durchzuführen. Gleichzeitig ist es jedoch wichtig, die Qualität der angebotenen Dienste zu verbessern, um neue Kunden zu gewinnen und langfristig an sich zu binden. Die mathematische Optimierung bietet sich für viele solcher Aufgabenstellungen als hervorragend geeignetes Planungswerkzeug an. Ziel dieses Artikels ist es, ihre Methodik und ihre Anwendung speziell zur Kosten- und Qualitätsoptimierung in Kommunikationsnetzen vorzustellen. Anhand von vier konkreten Planungsaufgaben aus dem Bereich der Festnetzplanung wird aufgezeigt, wie sich komplexe Zusammenhänge in flexiblen mathematischen Modellen abbilden lassen und welche Verfahren zur automatisierten Bearbeitung der Probleme eingesetzt werden können. Die hier vorgestellten Methoden zeichnen sich insbesondere dadurch aus, dass sie neben hochwertigen Lösungen auch eine Qualittsgarantie liefern, mit der sich die Lsungen fundiert bewerten lassen. Die dokumentierten Ergebnisse aus verschiedenen Industrieprojekten belegen die Eignung und Güte der mathematischen Optimierung für die Praxis. T3 - ZIB-Report - 03-31 Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7537 ER - TY - GEN A1 - Bley, Andreas A1 - Zymolka, Adrian T1 - Planung kostenoptimaler Informations- und Kommunikations-Infrastrukturen N2 - Beim Entwurf und Ausbau von Informations- und Kommunikationsnetzwerken m{ü}ssen zahlreiche interdependente Entscheidungen getroffen und gleichzeitig mannigfaltige Bedingungen ber{ü}cksichtigt werden. Die verf{ü}gbaren technischen und organisatorischen Alternativm{ö}glichkeiten sind normalerweise so vielf{ä}ltig und komplex, dass eine manuelle Planung praktisch nicht m{ö}glich ist. In diesem Artikel wird das Potential und die Methodik der mathematischen Optimierung bei der kostenoptimalen Planung von Kommunikationsnetzen vorgestellt. Als Ausgangspunkt wird exemplarisch eine typische praktische Aufgabe, die Struktur- und Konfigurationsplanung mehrstufiger Telekommunikationsnetzwerke, dargestellt. Anschließend werden kurz die wesentlichen Modellierungstechniken und Verfahrensans{ä}tze der mathematischen Optimierung skizziert. Abschließend gehen wir auf die Planung einer ad{ä}quaten Informations- und Kommunikations- Infrastruktur f{ü}r ein dezentrales Energieversorgungsnetz ein. T3 - ZIB-Report - 03-37 Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7591 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 - 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 - 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 - Bley, Andreas A1 - Koch, Thorsten T1 - Optimierung in der Planung und beim Aufbau des G-WiN N2 - Ende Juni diesen Jahres wurde das Gigabit-Wissenschaftsnetz offiziell gestartet. In der zweijährigen Vorbereitungsphase wurden nicht nur die technischen Möglichkeiten der neuen Übertragungstechniken und Dienste getestet. Es wurden auch verschiedene Fragestellungen zum effizienten Einsatz der verfügbaren Ressourcen für den Betrieb des G-WiN untersucht. In diesem Artikel beschreiben wir, wie das G-WiN zu seiner jetzigen Struktur und Topologie gekommen ist. T3 - ZIB-Report - 00-48 Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6169 ER -