The Gilbert Arborescence Problem

We investigate the problem of designing a minimum cost flow network interconnecting n sources and a single sink, each with known locations in a normed space and with associated flow demands. The network may contain any finite number of additional unprescribed nodes from the space; these are known as the Steiner points. For concave increasing cost functions, a minimum cost network of this sort has a tree topology, and hence can be called a Minimum Gilbert Arborescence (MGA). We characterise the local topological structure of Steiner points in MGAs, showing, in particular, that for a wide range of metrics, and for some typical real-world cost-functions, the degree of each Steiner point is 3.Comment: 19 pages, 7 figures. arXiv admin note: text overlap with arXiv:0903.212

Designing and Expanding Electrical Networks – Complexity and Combinatorial Algorithms

The transition from conventional to renewable power generation has a large impact on when and where electricity is generated. To deal with this change the electric transmission network needs to be adapted and expanded. Expanding the network has two benefits. Electricity can be generated at locations with high renewable energy potentials and then transmitted to the consumers via the transmission network. Without the expansion the existing transmission network may be unable to cope with the transmission needs, thus requiring power generation at locations closer to the energy demand, but at less well-suited locations. Second, renewable energy generation (e.g., from wind or solar irradiation) is typically volatile. Having strong interconnections between regions within a large geographical area allows to the smooth the generation and demand over that area. This smoothing makes them more predictable and the volatility of the generation easier to handle. In this thesis we consider problems that arise when designing and expanding electric transmission networks. As the first step we formalize them such that we have a precise mathematical problem formulation. Afterwards, we pursue two goals: first, improve the theoretical understanding of these problems by determining their computational complexity under various restrictions, and second, develop algorithms that can solve these problems. A basic formulation of the expansion planning problem models the network as a graph and potential new transmission lines as edges that may be added to the graph. We formalize this formulation as the problems Flow Expansion and Electrical Flow Expansion, which differ in the flow model (graph-theoretical vs. electrical flow). We prove that in general the decision variants of these problems are $\mathcal{NP}$-complete, even if the network structure is already very simple, e.g., a star. For certain restrictions, we give polynomial-time algorithms as well. Our results delineate the boundary between the $\mathcal{NP}$-complete cases and the cases that can be solved in polynomial time. The basic expansion planning problems mentioned above ignore that real transmission networks should still be able to operate if a small part of the transmission equipment fails. We employ a criticality measure from the literature, which measures the dynamic effects of the failure of a single transmission line on the whole transmission network. In a first step, we compare this criticality measure to the well-used $N-1$ criterion. Moreover, we formulate this criticality measure as a set of linear inequalities, which may be added to any formulation of a network design problem as a mathematical program. To exemplify this usage, we introduce the criticality criterion in two transmission network expansion planning problems, which can be formulated as mixed-integer linear programs (MILPs). We then evaluate the performance of solving the MILPs. Finally, we develop a greedy heuristic for one of the two problems, and compare its performance to solving the MILP. Microgrids play an important role in the electrification of rural areas. We formalize the design of the cable layout of a microgrid as a geometric optimization problem, which we call Microgrid Cable Layout. A key difference to the network design problems above is that there is no graph with candidate edges given. Instead, edges and new vertices may be placed anywhere in the plane. We present a hybrid genetic algorithm for Microgrid Cable Layout and evaluate it on a set of benchmark instances, which include a real microgrid in the Democratic Republic of the Congo. Finally, instead of expanding electrical networks one may place electric equipment such as FACTS (flexible AC transmission system). These influence the properties of the transmission lines such that the network can be used more efficiently. We apply a model of FACTS from the literature and study the problem whether a given network with given positions and properties of the FACTS admits an electrical flow provided that FACTS are set appropriately. We call such a flow a FACTS flow. In this thesis we prove that in general it is $\mathcal{NP}$-complete to determine whether a network admits a FACTS flow, and we present polynomial-time algorithms for two restricted cases

Algorithms for cartographic visualization

Maps are effective tools for communicating information to the general public and help people to make decisions in, for example, navigation, spatial planning and politics. The mapmaker chooses the details to put on a map and the symbols to represent them. Not all details need to be geographic: thematic maps, which depict a single theme or attribute, such as population, income, crime rate, or migration, can very effectively communicate the spatial distribution of the visualized attribute. The vast amount of data currently available makes it infeasible to design all maps manually, and calls for automated cartography. In this thesis we presented efficient algorithms for the automated construction of various types of thematic maps. In Chapter 2 we studied the problem of drawing schematic maps. Schematic maps are a well-known cartographic tool; they visualize a set of nodes and edges (for example, highway or metro networks) in simplified form to communicate connectivity information as effectively as possible. Many schematic maps deviate substantially from the underlying geography since edges and vertices of the original network are moved in the simplification process. This can be a problem if we want to integrate the schematized network with a geographic map. In this scenario the schematized network has to be drawn with few orientations and links, while critical features (cities, lakes, etc.) of the base map are not obscured and retain their correct topological position with respect to the network. We developed an efficient algorithm to compute a collection of non-crossing paths with fixed orientations using as few links as possible. This algorithm approximates the optimal solution to within a factor that depends only on the number of allowed orientations. We can also draw the roads with different thicknesses, allowing us to visualize additional data related to the roads such as trafic volume. In Chapter 3 we studied methods to visualize quantitative data related to geographic regions. We first considered rectangular cartograms. Rectangular cartograms represent regions by rectangles; the positioning and adjacencies of these rectangles are chosen to suggest their geographic locations to the viewer, while their areas are chosen to represent the numeric values being communicated by the cartogram. One drawback of rectangular cartograms is that not every rectangular layout can be used to visualize all possible area assignments. Rectangular layouts that do have this property are called area-universal. We show that area-universal layouts are always one-sided, and we present algorithms to find one-sided layouts given a set of adjacencies. Rectangular cartograms often provide a nice visualization of quantitative data, but cartograms deform the underlying regions according to the data, which can make the map virtually unrecognizable if the data value differs greatly from the original area of a region or if data is not available at all for a particular region. A more direct method to visualize the data is to place circular symbols on the corresponding region, where the areas of the symbols correspond to the data. However, these maps, so-called symbol maps, can appear very cluttered with many overlapping symbols if large data values are associated with small regions. In Chapter 4 we proposed a novel type of quantitative thematic map, called necklace map, which overcomes these limitations. Instead of placing the symbols directly on a region, we place the symbols on a closed curve, the necklace, which surrounds the map. The location of a symbol on the necklace should be chosen in such a way that the relation between symbol and region is as clear as possible. Necklace maps appear clear and uncluttered and allow for comparatively large symbol sizes. We developed algorithms to compute necklace maps and demonstrated our method with experiments using various data sets and maps. In Chapter 5 and 6 we studied the automated creation of ow maps. Flow maps are thematic maps that visualize the movement of objects, such as people or goods, between geographic regions. One or more sources are connected to several targets by lines whose thickness corresponds to the amount of ow between a source and a target. Good ow maps reduce visual clutter by merging (bundling) lines smoothly and by avoiding self-intersections. We developed a new algorithm for drawing ow trees, ow maps with a single source. Unlike existing methods, our method merges lines smoothly and avoids self-intersections. Our method is based on spiral trees, a new type of Steiner trees that we introduced. Spiral trees have an angle restriction which makes them appear smooth and hence suitable for drawing ow maps. We study the properties of spiral trees and give an approximation algorithm to compute them. We also show how to compute ow trees from spiral trees and we demonstrate our approach with extensive experiments

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

Well-solvable special cases of the TSP : a survey

The Traveling Salesman Problem belongs to the most important and most investigated problems in combinatorial optimization. Although it is an NP-hard problem, many of its special cases can be solved efficiently. We survey these special cases with emphasis on results obtained during the decade 1985-1995. This survey complements an earlier survey from 1985 compiled by Gilmore, Lawler and Shmoys. Keywords: Traveling Salesman Problem, Combinatorial optimization, Polynomial time algorithm, Computational complexity

Mathematical Imaging and Surface Processing

Within the last decade image and geometry processing have become increasingly rigorous with solid foundations in mathematics. Both areas are research fields at the intersection of different mathematical disciplines, ranging from geometry and calculus of variations to PDE analysis and numerical analysis. The workshop brought together scientists from all these areas and a fruitful interplay took place. There was a lively exchange of ideas between geometry and image processing applications areas, characterized in a number of ways in this workshop. For example, optimal transport, first applied in computer vision is now used to define a distance measure between 3d shapes, spectral analysis as a tool in image processing can be applied in surface classification and matching, and so on. We have also seen the use of Riemannian geometry as a powerful tool to improve the analysis of multivalued images. This volume collects the abstracts for all the presentations covering this wide spectrum of tools and application domains

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor