Notes on large angle crossing graphs

A graph G is an a-angle crossing (aAC) graph if every pair of crossing edges in G intersect at an angle of at least a. The concept of right angle crossing (RAC) graphs (a=Pi/2) was recently introduced by Didimo et. al. It was shown that any RAC graph with n vertices has at most 4n-10 edges and that there are infinitely many values of n for which there exists a RAC graph with n vertices and 4n-10 edges. In this paper, we give upper and lower bounds for the number of edges in aAC graphs for all 0 < a < Pi/2

Computational Aspects of Treewidth -- Lower Bounds and Network Reliability

Good treewidth lower bounds can be used in branch-and-bound methods. The better and faster the bounds, the faster the branch-and-bound algorithm. They are also useful to estimate the running time of a dynamic programming algorithm based on tree-decompositions. A large treewidth lower bound indicates that the treewidth is large. Because of the exponential influence of the treewidth on the running time of such methods, there is only little hope to find an efficient algorithm based on tree-decompositions. Treewidth lower bounds in connection with treewidth upper bounds can help to assess the quality of these bounds. A small gap between the bounds means tight bounds, and a large gap indicates room for improvements. Communication networks are an important part of our world. Real networks consist of elements that are not infallible. They can be modelled by graphs, where we associate to each vertex a rational number that is the reliability of the corresponding network element. We assume that the reliabilities of the elements are stochastically independent. A very crucial issue for designing and maintaining networks is their reliability. The notion network reliability addresses questions such as: 'What is the probability that two distinguished sites can communicate, while parts of the network broke down?' Computing the network reliability is in general NP-hard. In this thesis, we develop new treewidth lower bounds. All new lower bounds have in common that they are based on a combination of existing (degree-based) lower bounds and edge contractions. Contracting an edge is the operation that replaces an edge and its two endpoints by one vertex that is made adjacent to all the neighbours of the two endpoint of the contracted edge. A minor of a graph is a graph obtained from a subgraph by contracting edges. The main idea to improve treewidth lower bounds is to take an existing lower bound over all subgraphs or minors. In this way, we obtain a number of parameters, all treewidth lower bounds, study relations between them and their computational complexity. For the parameters that are NP-hard to compute we develop heuristics. In experiments, we compare the quality of the lower bounds and their running times amongst each other and also to the best known treewidth upper bounds for a number of graphs from various areas such as probabilistic networks or frequency assignment problems. The results of the experiments made very clear that combining edge contraction with existing treewidth lower bounds is a very vital idea to improve upon treewidth lower bounds. One treewidth lower bound parameter is the contraction degeneracy. It is the maximum over all minors of a graph of the minimum degree of the minor. This parameter is NP-hard to compute. Due to its elementary character, it is an interesting study object in its own right and not only as a treewidth lower bound. We present a polynomial time method for computing the contraction degeneracy based on dynamic programming for cographs. We also present a framework for network reliability problems for graphs of bounded treewidth. In our model we have two distinguished sets of vertices (severs and clients). With the framework we can answer more questions, such as 'What is the probability that every client is connected to at least one server?' or 'What is the expected number of connected components with at least one server?'. These and similar question are proven to be #P-hard. However, using our framework they can be solved in polynomial time on graphs of bounded treewidth

A note on the complexity of network reliability problems,” downloadable from http://www.cs.uu.nl/research/techreps/aut/thomasw.html

Abstract. Let be given an undirected, simple graph G = (V, E). We associate to each vertex a number in [0, 1]- its reliability, i.e. the probability that it does not fail. Furthermore, let a set S ⊆ V of servers and a set L ⊆ V of clients be given. Vertex failures are independent of each other. The network reliability asks for the probability that the graph induced by the non-failed vertices is connected. We can also ask for the probabilities of connections between vertices of S and between vertices of S and L, e.g. ‘What is the probability that all clients are connected to at least one server?’ In this note, we consider a list of such problems and prove their #P-completeness.

Football analysis using spatio-temporal tools

Analysing a football match is without doubt an important task for coaches, talent scouts, players and even media; and with current technologies more and more match data is collected. Several companies offer the ability to track the position of the players and the ball with high accuracy and high resolution. They also offer software that include basic analysis tools, for example straight-forward statistics about distance run and number of passes. It is, however, a non-trivial task to perform more advanced analysis. We present a collection of tools that we developed specifically for analysing the performance of football players and teams. The aim, functionality and the underlying algorithms for each tool are presented and discussed. 1

Contraction Degeneracy on Cographs

The contraction degeneracy of a graph G is the maximum minimum degree of G # over all minors G # of G. The corresponding decision problem is known to be NP-complete. In thi

A Note on Edge Contraction

Contracting an edge is the operation that introduces a new vertex that is adjacent to all vertices the endpoints of the contracted edge are adjacent to, and then deletes the endpoints of this edge and all their incident edges. In this note, we give a formal approach to the notion of edge contraction and show some basic properties of it

Computational movement analysis

Recent advances in tracking technologies result in geographic information representing the movement of individuals at previously unseen spatial and temporal granularities. This new, inherently spatiotemporal, kind of geographic information offers new insights into dynamic geographic processes but also challenges the traditionally rather static spatial analysis toolbox. This chapter presents an introductory overview to movement data in general, the theory for modeling and analyzing movement, as well as a set of key application fields of movement analysis. Finally, the chapter addresses privacy concerns relevant to the analysis of human movement

Computer-gestützte Bewegungsanalyse

Die jüngsten Fortschritte der Trackingtechnologie produzieren Geodaten, welche die Bewegung mobiler Objekte mit einer bisher unerreichten räumlichen und zeitlichen Auflösung erfassen. Diese neue, von Natur aus raumzeitliche Art geographischer Informationen ermöglicht neue Einsichten in dynamische geographische Prozesse, stellt aber auch die traditionell eher statischen Werkzeuge der Raumanalyse infrage. Dieses Kapitel gibt einen Überblick über Bewegungsdaten im Allgemeinen, die Theorie der Bewegungsmodellierung und -analyse sowie eine Reihe wichtiger Anwendungsfelder der computer-gestützten Bewegungsanalyse. Schließlich geht das Kapitel auf Überlegungen bezüglich der Privatsphäre ein, welche für die Analyse der Bewegung von Menschen sehr wichtig sind