90C35 Programming involving graphs or networks [See also 90C27]
Refine
Year of publication
Document Type
- ZIB-Report (26)
- Article (2)
- Doctoral Thesis (1)
Is part of the Bibliography
- no (29)
Keywords
- integer programming (3)
- network design (3)
- Pooling Problem (2)
- Relaxation (2)
- Steiner tree packing (2)
- discrete optimization (2)
- discrete-continuous algorithm (2)
- flight planning (2)
- free flight (2)
- mixed integer programming (2)
Institute
- Mathematical Optimization (12)
- ZIB Allgemein (12)
- Network Optimization (6)
- Energy Network Optimization (2)
- Mathematical Algorithmic Intelligence (2)
- Mathematics for Life and Materials Science (2)
- Mathematics of Transportation and Logistics (2)
- Modeling and Simulation of Complex Processes (2)
- Computational Medicine (1)
- Mathematics of Health Care (1)
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.
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.
We study the Flight Planning Problem for a single aircraft, where we look for a minimum cost path in the airway network, a directed graph. Arc evaluation, such as weather computation, is computationally expensive due to non-linear functions, but required for exactness. We propose several pruning methods to thin out the search space for Dijkstra's algorithm before the query commences. We do so by using innate problem characteristics such as an aircraft's tank capacity, lower and upper bounds on the total costs, and in particular, we present a method to reduce the search space even in the presence of regional crossing costs.
We test all pruning methods on real-world instances, and show that incorporating crossing costs into the pruning process can reduce the number of nodes by 90\% in our setting.
We propose in this paper the Dynamic Multiobjective Shortest Problem. It features multidimensional states that can depend on several variables and not only on time; this setting is motivated by flight planning and electric vehicle routing applications. We give an exact algorithm for the FIFO case and derive from it an FPTAS, which is computationally efficient. It also features the best known complexity in the static case.
Let $G$ be a directed acyclic graph with $n$ arcs, a source $s$ and a sink $t$. We introduce the cone $K$ of flow matrices, which is a polyhedral cone
generated by the matrices $1_P 1_P^T \in R^{n\times n}$, where
$1_P\in R^n$ is the incidence vector of the $(s,t)$-path $P$.
Several combinatorial problems reduce to a linear optimization problem over $K$.
This cone is intractable, but we provide two convergent approximation hierarchies, one of them based on a
completely positive representation of $K$.
We illustrate this approach by computing bounds for a maximum flow problem with pairwise arc-capacities.
This paper deals with directed, bidirected, and undirected capacitated network design problems. Using mixed integer rounding (MIR), we generalize flow-cutset inequalities to these three link types and to an arbitrary modular link capacity structure, and propose a generic separation algorithm. In an extensive computational study on 54 instances from the Survivable Network Design Library (SNDlib), we show that the performance of cplex can significantly be enhanced by this class of cutting planes. The computations reveal the particular importance of the subclass of cutset-inequalities.
Abstract The cost-efficient design of survivable optical telecommunication networks is the topic of this thesis. In cooperation with network operators, we have developed suitable concepts and mathematical optimization methods to solve this comprehensive planning task in practice. Optical technology is more and more employed in modern telecommunication networks. Digital information is thereby transmitted as short light pulses through glass fibers. Moreover, the optical medium allows for simultaneous transmissions on a single fiber by use of different wavelengths. Recent optical switches enable a direct forwarding of optical channels in the network nodes without the previously required signal retransformation to electronics. Their integration creates ongoing optical connections,which are called lightpaths. We study the problem of finding cost-efficient configurations of optical networks which meet specified communication requirements. A configuration comprises the determination of all lightpaths to establish as well as the detailed allocation of all required devices and systems. We use a flexible modeling framework for a realistic representation of the networks and their composition. For different network architectures, we formulate integer linear programs which model the design task in detail. Moreover, network survivability is an important issue due to the immense bandwidths offered by optical technology. Operators therefore request for designs which perpetuate protected connections and guarantee for a defined minimum throughput in case of malfunctions. In order to achieve an effective realization of scalable protection, we present a novel survivability concept tailored to optical networks and integrate several variants into the models. Our solution approach is based on a suitable model decomposition into two subtasks which separates two individually hard subproblems and enables this way to compute cost-efficient designs with approved quality guarantee. The first subtask consists of routing the connections with corresponding dimensioning of capacities and constitutes a common core task in the area of network planning. Sophisticated methods for such problems have already been developed and are deployed by appropriate integration. The second subtask is characteristic for optical networks and seeks for a conflict-free assignment of available wavelengths to the lightpaths using a minimum number of involved wavelength converters. For this coloring-like task, we derive particular models and study methods to estimate the number of unavoidable conversions. As constructive approach, we develop heuristics and an exact branch-and-price algorithm. Finally, we carry out an extensive computational study on realistic data, provided by our industrial partners. As twofold purpose, we demonstrate the potential of our approach for computing good solutions with quality guarantee, and we exemplify its flexibility for application to network design and analysis.
Orbitopes can be used to handle symmetries which arise in integer programming formulations with an inherent assignment structure. We investigate the detection of symmetries appearing in this approach. We show that detecting so-called orbitopal symmetries is graph-isomorphism hard in general, but can be performed in linear time if the assignment structure is known.
We estimate potential energy savings in IP-over-WDM networks achieved by switching off router line cards in low-demand hours. We compare three approaches to react on dynamics in the IP traffic over time, FUFL, DUFL and DUDL. They provide different levels of freedom in adjusting the routing of lightpaths in the WDM layer and the routing of demands in the IP layer. Using MILP models based on three realistic network topologies as well as realistic demands, power, and cost values, we show that already a simple monitoring of the lightpath utilization in order to deactivate empty line cards (FUFL) brings substantial benefits. The most significant savings, however, are achieved by rerouting traffic in the IP layer (DUFL), which allows emptying and deactivating lightpaths together with the corresponding line cards. A sophisticated reoptimization of the virtual topologies and the routing in the optical domain for every demand scenario (DUDL) yields nearly no additional profits in the considered networks. These results are independent of the ratio between the demand and capacity granularities, the time scale and the network topology, and show little dependency on the demand structure.
In this article, strategical infrastructure planning problems in the design of large-scale telecommunication networks are discussed based on experiences from three projects with industrial partners: The access network planning of the German Gigabit-Wissenschaftsnetz (G-WiN) for DFN (Verein zur Förderung eines Deutschen Forschungsnetzes e.V.), the mobile network switching center location planning project for E-Plus Mobilfunk, and the fixed network switching center location planning project for TELEKOM AUSTRIA. We introduce a mathematical model for a hierarchical multi-commodity capacitated facility location problem, present adaptions of this basic model to the specific requirements within the different projects and discuss the individual peculiarities and model decisions made. Eventually, we present and discuss computational results of three associated case studies, illustrating '"how we did the job`` with mathematical methods.