Approximation Algorithms for Network Design Problems

We consider different variants of network design problems. Given a set of points in the plane we search for a shortest interconnection of them. In this general formulation the problem is known as Steiner tree problem. We consider the special case of octilinear Steiner trees in the presence of hard and soft obstacles. In an octilinear Steiner tree the line segments connecting the points are allowed to run either in horizontal, vertical or diagonal direction. An obstacle is a connected region in the plane bounded by a simple polygon. No line segment of an octilinear Steiner tree is allowed to lie in the interior of a hard obstacle. If we intersect a Steiner tree with the interior of a soft obstacle, no connected component of the induced subtree is allowed to be longer than a given fixed length. We provide polynomial time approximation schemes for the octilinear Steiner tree problem in the presence of hard and soft obstacles. These were the first presented approximation schemes introduced for the problems. Additionally, we introduce a (2+epsilon)-approximation algorithm for soft obstacles. We then turn to Euclidean group Steiner trees. Instead of a set of fixed points we get for each point a set of potential locations (combined into groups) and we need to pick only one location of each group. The groups we consider lie inside disjoint regions fulfilling a special property so-called alpha-fatness. Roughly speaking, the term alpha-fat specifies the shape of the region in comparison to a disk. We give the first approximation algorithm for this problem and achieve an approximation ratio of (1+epsilon)(9.093alpha +1). Last, we consider Manhattan networks. They are allowed to contain edges only in horizontal and vertical direction. In contrast to Steiner trees they contain a shortest path between each pair of points. We introduce insights into the structure of Manhattan networks, particularly in the context of so-called staircases. We give three new approximation algorithms for the Manhattan network problem, the first with approximation ratio 3 and two algorithms with ratio 2. To this end we introduce two algorithms for the Manhattan network problem of staircases. The first algorithm solves the problem to optimality the second yields a 2-approximation. Variants of both algorithms are already known in the literature. Since we use a slightly different definition of staircases and we need special properties of them, we adopt the algorithms to our situation. The 2-approximation algorithms achieve the best known approximation ratio of an algorithm for the Manhattan network problem known so far. Last we give an idea how we could possibly find an algorithm with better approximation ratio

Combinatorial Optimization

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

Abstracts for the twentyfirst European workshop on Computational geometry, Technische Universiteit Eindhoven, The Netherlands, March 9-11, 2005

This volume contains abstracts of the papers presented at the 21st European Workshop on Computational Geometry, held at TU Eindhoven (the Netherlands) on March 9–11, 2005. There were 53 papers presented at the Workshop, covering a wide range of topics. This record number shows that the field of computational geometry is very much alive in Europe. We wish to thank all the authors who submitted papers and presented their work at the workshop. We believe that this has lead to a collection of very interesting abstracts that are both enjoyable and informative for the reader. Finally, we are grateful to TU Eindhoven for their support in organizing the workshop and to the Netherlands Organisation for Scientific Research (NWO) for sponsoring the workshop

The Homogeneous Broadcast Problem in Narrow and Wide Strips

Let $P$ be a set of nodes in a wireless network, where each node is modeled as a point in the plane, and let $s\in P$ be a given source node. Each node $p$ can transmit information to all other nodes within unit distance, provided $p$ is activated. The (homogeneous) broadcast problem is to activate a minimum number of nodes such that in the resulting directed communication graph, the source $s$ can reach any other node. We study the complexity of the regular and the hop-bounded version of the problem (in the latter, $s$ must be able to reach every node within a specified number of hops), with the restriction that all points lie inside a strip of width $w$. We almost completely characterize the complexity of both the regular and the hop-bounded versions as a function of the strip width $w$.Comment: 50 pages, WADS 2017 submissio

Decomposition of Geometric Set Systems and Graphs

We study two decomposition problems in combinatorial geometry. The first part deals with the decomposition of multiple coverings of the plane. We say that a planar set is cover-decomposable if there is a constant m such that any m-fold covering of the plane with its translates is decomposable into two disjoint coverings of the whole plane. Pach conjectured that every convex set is cover-decomposable. We verify his conjecture for polygons. Moreover, if m is large enough, we prove that any m-fold covering can even be decomposed into k coverings. Then we show that the situation is exactly the opposite in 3 dimensions, for any polyhedron and any $m$ we construct an m-fold covering of the space that is not decomposable. We also give constructions that show that concave polygons are usually not cover-decomposable. We start the first part with a detailed survey of all results on the cover-decomposability of polygons. The second part investigates another geometric partition problem, related to planar representation of graphs. The slope number of a graph G is the smallest number s with the property that G has a straight-line drawing with edges of at most s distinct slopes and with no bends. We examine the slope number of bounded degree graphs. Our main results are that if the maximum degree is at least 5, then the slope number tends to infinity as the number of vertices grows but every graph with maximum degree at most 3 can be embedded with only five slopes. We also prove that such an embedding exists for the related notion called slope parameter. Finally, we study the planar slope number, defined only for planar graphs as the smallest number s with the property that the graph has a straight-line drawing in the plane without any crossings such that the edges are segments of only s distinct slopes. We show that the planar slope number of planar graphs with bounded degree is bounded.Comment: This is my PhD thesi