Dominators in Directed Graphs: A Survey of Recent Results, Applications, and Open Problems

The computation of dominators is a central tool in program optimization and code generation, and it has applications in other diverse areas includingconstraint programming, circuit testing, and biology. In this paper we survey recent results, applications, and open problems related to the notion of dominators in directed graphs,including dominator verification and certification, computing independent spanning trees, and connectivity and path-determination problems in directed graphs

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures

A discrete model for studying existence and uniqueness of solutions in nonlinear resistive circuits

AbstractTwo combinatorial problems raised by the fundamental question of the existence and uniqueness of solutions in nonlinear electric circuits are presented. The first problem, namely the existence of a pair of conjugate trees, has been solved in polynomial time using an original model based on matroïd intersection. For the second problem, which is the search of a particular orientation in a multigraph with labeled edges, an elaborate branch and bound procedure is proposed

Paths and walks, forests and planes : arcadian algorithms and complexity

This dissertation is concerned with new results in the area of parameterized algorithms and complexity. We develop a new technique for hard graph problems that generalizes and unifies established methods such as Color-Coding, representative families, labelled walks and algebraic fingerprinting. At the heart of the approach lies an algebraic formulation of the problems, which is effected by means of a suitable exterior algebra. This allows us to estimate the number of simple paths of given length in directed graphs faster than before. Additionally, we give fast deterministic algorithms for finding paths of given length if the input graph contains only few of such paths. Moreover, we develop faster deterministic algorithms to find spanning trees with few leaves. We also consider the algebraic foundations of our new method. Additionally, we investigate the fine-grained complexity of determining the precise number of forests with a given number of edges in a given undirected graph. To wit, this happens in two ways. Firstly, we complete the complexity classification of the Tutte plane, assuming the exponential time hypothesis. Secondly, we prove that counting forests with a given number of edges is at least as hard as counting cliques of a given size.Diese Dissertation befasst sich mit neuen Ergebnissen auf dem Gebiet parametrisierter Algorithmen und Komplexitätstheorie. Wir entwickeln eine neue Technik für schwere Graphprobleme, die etablierte Methoden wie Color-Coding, representative families, labelled walks oder algebraic fingerprinting verallgemeinert und vereinheitlicht. Kern der Herangehensweise ist eine algebraische Formulierung der Probleme, die vermittels passender Graßmannalgebren geschieht. Das erlaubt uns, die Anzahl einfacher Pfade gegebener Länge in gerichteten Graphen schneller als bisher zu schätzen. Außerdem geben wir schnelle deterministische Verfahren an, Pfade gegebener Länge zu finden, falls der Eingabegraph nur wenige solche Pfade enthält. Übrigens entwickeln wir schnellere deterministische Algorithmen, um Spannbäume mit wenigen Blättern zu finden. Wir studieren außerdem die algebraischen Grundlagen unserer neuen Methode. Weiters untersuchen wir die fine-grained-Komplexität davon, die genaue Anzahl von Wäldern einer gegebenen Kantenzahl in einem gegebenen ungerichteten Graphen zu bestimmen. Und zwar erfolgt das auf zwei verschiedene Arten. Erstens vervollständigen wir die Komplexitätsklassifizierung der Tutte-Ebene unter Annahme der Expo- nentialzeithypothese. Zweitens beweisen wir, dass Wälder mit gegebener Kantenzahl zu zählen, wenigstens so schwer ist, wie Cliquen gegebener Größe zu zählen.Cluster of Excellence (Multimodal Computing and Interaction

An efficient parallel algorithm for planarity

AbstractWe describe a parallel algorithm for testing a graph for planarity, and for finding an embedding of a planar graph. For a graph on n vertices, the algorithm runs in O(log2n) steps on n processors of a parallel RAM. The previous best parallel algorithm for planarity testing also ran in O(log2n) time (J. Ja'Ja' and J. Simon, J Comput.11, No. 2 (1982), 313–328), but used a reduction to solving linear systems, and hence required Ω(M(n)log2n) processors, where M(n) is the sequential time for n × n matrix multiplication, whereas our processor bounds are within a polylog factor of optimal. The most significant aspect of our parallel algorithms is the use of a sophisticated data structure for representing sets of embeddings, the PQ-tree of K. Booth and G. Lueker, J. Comput. System Sci.13, No. 3 (1976), 335–379). Previously no parallel algorithms for PQ-trees were known. We have efficient parallel algorithms for manipulating PQ-trees, which we use in our planarity algorithm

Hierarchical Graphs as Organisational Principle and Spatial Model Applied to Pedestrian Indoor Navigation

In this thesis, hierarchical graphs are investigated from two different angles – as a general modelling principle for (geo)spatial networks and as a practical means to enhance navigation in buildings. The topics addressed are of interest from a multi-disciplinary point of view, ranging from Computer Science in general over Artificial Intelligence and Computational Geometry in particular to other fields such as Geographic Information Science. Some hierarchical graph models have been previously proposed by the research community, e.g. to cope with the massive size of road networks, or as a conceptual model for human wayfinding. However, there has not yet been a comprehensive, systematic approach for modelling spatial networks with hierarchical graphs. One particular problem is the gap between conceptual models and models which can be readily used in practice. Geospatial data is commonly modelled - if at all - only as a flat graph. Therefore, from a practical point of view, it is important to address the automatic construction of a graph hierarchy based on the predominant data models. The work presented deals with this problem: an automated method for construction is introduced and explained. A particular contribution of my thesis is the proposition to use hierarchical graphs as the basis for an extensible, flexible architecture for modelling various (geo)spatial networks. The proposed approach complements classical graph models very well in the sense that their expressiveness is extended: various graphs originating from different sources can be integrated into a comprehensive, multi-level model. This more sophisticated kind of architecture allows for extending navigation services beyond the borders of one single spatial network to a collection of heterogeneous networks, thus establishing a meta-navigation service. Another point of discussion is the impact of the hierarchy and distribution on graph algorithms. They have to be adapted to properly operate on multi-level hierarchies. By investigating indoor navigation problems in particular, the guiding principles are demonstrated for modelling networks at multiple levels of detail. Complex environments like large public buildings are ideally suited to demonstrate the versatile use of hierarchical graphs and thus to highlight the benefits of the hierarchical approach. Starting from a collection of floor plans, I have developed a systematic method for constructing a multi-level graph hierarchy. The nature of indoor environments, especially their inherent diversity, poses an additional challenge: among others, one must deal with complex, irregular, and/or three-dimensional features. The proposed method is also motivated by practical considerations, such as not only finding shortest/fastest paths across rooms and floors, but also by providing descriptions for these paths which are easily understood by people. Beyond this, two novel aspects of using a hierarchy are discussed: one as an informed heuristic exploiting the specific characteristics of indoor environments in order to enhance classical, general-purpose graph search techniques. At the same time, as a convenient by- product of this method, clusters such as sections and wings can be detected. The other reason is to better deal with irregular, complex-shaped regions in a way that instructions can also be provided for these spaces. Previous approaches have not considered this problem. In summary, the main results of this work are: • hierarchical graphs are introduced as a general spatial data infrastructure. In particular, this architecture allows us to integrate different spatial networks originating from different sources. A small but useful set of operations is proposed for integrating these networks. In order to work in a hierarchical model, classical graph algorithms are generalised. This finding also has implications on the possible integration of separate navigation services and systems; • a novel set of core data structures and algorithms have been devised for modelling indoor environments. They cater to the unique characteristics of these environments and can be specifically used to provide enhanced navigation in buildings. Tested on models of several real buildings from our university, some preliminary but promising results were gained from a prototypical implementation and its application on the models

Exploiting graph structures for computational efficiency

Coping with NP-hard graph problems by doing better than simply brute force is a field of significant practical importance, and which have also sparked wide theoretical interest. One route to cope with such hard graph problems is to exploit structures which can possibly be found in the input data or in the witness for a solution. In the framework of parameterized complexity, we attempt to quantify such structures by defining numbers which describe "how structured" the graph is. We then do a fine-grained classification of its computational complexity, where not only the input size, but also the structural measure in question come in to play. There is a number of structural measures called width parameters, which includes treewidth, clique-width, and mim-width. These width parameters can be compared by how many classes of graphs that have bounded width. In general there is a tradeoff; if more graph classes have bounded width, then fewer problems can be efficiently solved with the aid of a small width; and if a width is bounded for only a few graph classes, then it is easier to design algorithms which exploit the structure described by the width parameter. For each of the mentioned width parameters, there are known meta-theorems describing algorithmic results for a wide array of graph problems. Hence, showing that decompositions with bounded width can be found for a certain graph class yields algorithmic results for the given class. In the current thesis, we show that several graph classes have bounded width measures, which thus gives algorithmic consequences. Algorithms which are FPT or XP parameterized by width parameters are exploiting structure of the input graph. However, it is also possible to exploit structures that are required of a witness to the solution. We use this perspective to give a handful of polynomial-time algorithms for NP-hard problems whenever the witness belongs to certain graph classes. It is also possible to combine structures of the input graph with structures of the solution witnesses in order to obtain parameterized algorithms, when each structure individually is provably insufficient to provide so under standard complexity assumptions. We give an example of this in the final chapter of the thesis