Kernel Bounds for Path and Cycle Problems

Connectivity problems like k-Path and k-Disjoint Paths relate to many important milestones in parameterized complexity, namely the Graph Minors Project, color coding, and the recent development of techniques for obtaining kernelization lower bounds. This work explores the existence of polynomial kernels for various path and cycle problems, by considering nonstandard parameterizations. We show polynomial kernels when the parameters are a given vertex cover, a modulator to a cluster graph, or a (promised) max leaf number. We obtain lower bounds via cross-composition, e.g., for Hamiltonian Cycle and related problems when parameterized by a modulator to an outerplanar graph

Improving the capacity of radio spectrum: exploration of the acyclic orientations of a graph

The efficient use of radio spectrum depends upon frequency assignment within a telecommunications network. The solution space of the frequency assignment problem is best described by the acyclic orientations of the network. An acyclic orientation Ɵ of a graph (network) G is an orientation of the edges of the graph which does not create any directed cycles. We are primarily interested in how many ways this is possible for a given graph, which is the count of the number of acyclic orientations, a(G). This is just the evaluation of the chromatic polynomial of the graph χ(G; λ) at λ = -1. Calculating (and even approximating) the chromatic polynomial is known to be #P-hard, but it is unknown whether or not the approximation at the value -1 is. There are two key contributions in this thesis. Firstly, we obtain computational results for all graphs with up to 8 vertices. We use the data to make observations on the structure of minimal and maximal graphs, by which we mean graphs with the fewest and greatest number of acyclic orientations respectively, as well as on the distribution of acyclic orientations. Many conjectures on the structure of extremal graphs arise, of which we prove some in the theoretical part of the thesis. Secondly, we present a compression move which is monotonic with respect to the number of acyclic orientations, and with respect to various other parameters in particular cliques. This move gives us a new approach to classifying all minimal graphs. It also enables us to tackle the harder problem of identifying maximal graphs. We show that certain Turán graphs are uniquely maximal (Turán graphs are complete multipartite graphs with all vertex classes as equal as possible), and conjecture that all Turán graphs are maximal. In addition we derive an explicit formula for the number of acyclic orientations of complete bipartite graphs

On graph algorithms for large-scale graphs

Die Anforderungen an Algorithmen hat sich in den letzten Jahren grundlegend geändert. Die Datengröße der zu verarbeitenden Daten wächst schneller als die zur Verfügung stehende Rechengeschwindigkeit. Daher sind neue Algorithmen auf sehr großen Graphen wie z.B. soziale Netzwerke, Computernetzwerke oder Zustandsübergangsgraphen entwickelt worden, um das Problem der immer größer werdenden Daten zu bewältigen. Diese Arbeit beschäftigt sich mit zwei Herangehensweisen für dieses Problem. Implizite Algorithmen benutzten eine verlustfreie Kompression der Daten, um die Datengröße zu reduzieren, und arbeiten direkt mit den komprimierten Daten, um Optimierungsprobleme zu lösen. Graphen werden hier anhand der charakteristischen Funktion der Kantenmenge dargestellt, welche mit Hilfe von Ordered Binary Decision Diagrams (OBDDs) – eine bekannte Datenstruktur für Boolesche Funktionen - repräsentiert werden können. Wir entwickeln in dieser Arbeit neue Techniken, um die OBDD-Größe von Graphen zu bestimmen, und wenden diese Technik für mehrere Klassen von Graphen an und erhalten damit (fast) optimale Schranken für die OBDD-Größen. Kleine Eingabe-OBDDs sind essenziell für eine schnelle Verarbeitung, aber wir brauchen auch Algorithmen, die große Zwischenergebnisse während der Ausführung vermeiden. Hierfür entwickeln wir Algorithmen für bestimme Graphklassen, die die Kodierung der Knoten ausnutzt, die wir für die Resultate der OBDD-Größe benutzt haben. Zusätzlich legen wir die Grundlage für die Betrachtung von randomisierten OBDD-basierten Algorithmen, indem wir untersuchen, welche Art von Zufall wir hier verwenden und wie wir damit Algorithmen entwerfen können. Im Zuge dessen geben wir zwei randomisierte Algorithmen an, die ihre entsprechenden deterministischen Algorithmen in einer experimentellen Auswertung überlegen sind. Datenstromalgoritmen sind eine weitere Möglichkeit für die Bearbeitung von großen Graphen. In diesem Modell wird der Graph anhand eines Datenstroms von Kanteneinfügungen repräsentiert und den Algorithmen steht nur eine begrenzte Menge von Speicher zur Verfügung. Lösungen für Graphoptimierungsprobleme benötigen häufig eine lineare Größe bzgl. der Anzahl der Knoten, was eine triviale untere Schranke für die Streamingalgorithmen für diese Probleme impliziert. Die Berechnung eines Matching ist so ein Beispiel, was aber in letzter Zeit viel Aufmerksamkeit in der Streaming-Community auf sich gezogen hat. Ein Matching ist eine Menge von Kanten, so dass keine zwei Kanten einen gemeinsamen Knoten besitzen. Wenn wir nur an der Größe oder dem Gewicht (im Falle von gewichteten Graphen) eines Matching interessiert sind, ist es mögliche diese lineare untere Schranke zu durchbrechen. Wir konzentrieren uns in dieser Arbeit auf dynamische Datenströme, wo auch Kanten gelöscht werden können. Wir reduzieren das Problem, einen Schätzer für ein gewichtsoptimales Matching zu finden, auf das Problem, die Größe von Matchings zu approximieren, wobei wir einen kleinen Verlust bzgl. der Approximationsgüte in Kauf nehmen müssen. Außerdem präsentieren wir den ersten dynamischen Streamingalgorithmus, der die Größe von Matchings in lokal spärlichen Graphen approximiert. Für kleine Approximationsfaktoren zeigen wir eine untere Schranke für den Platzbedarf von Streamingalgorithmen, die die Matchinggröße approximieren.The algorithmic challenges have changed in the last decade due to the rapid growth of the data set sizes that need to be processed. New types of algorithms on large graphs like social graphs, computer networks, or state transition graphs have emerged to overcome the problem of ever-increasing data sets. In this thesis, we investigate two approaches to this problem. Implicit algorithms utilize lossless compression of data to reduce the size and to directly work on this compressed representation to solve optimization problems. In the case of graphs we are dealing with the characteristic function of the edge set which can be represented by Ordered Binary Decision Diagrams (OBDDs), a well-known data structure for Boolean functions. We develop a new technique to prove upper and lower bounds on the size of OBDDs representing graphs and apply this technique to several graph classes to obtain (almost) optimal bounds. A small input OBDD size is absolutely essential for dealing with large graphs but we also need algorithms that avoid large intermediate results during the computation. For this purpose, we design algorithms for a specific graph class that exploit the encoding of the nodes that we use for the results on the OBDD sizes. In addition, we lay the foundation on the theory of randomization in OBDD-based algorithms by investigating what kind of randomness is feasible and how to design algorithms with it. As a result, we present two randomized algorithms that outperform known deterministic algorithms on many input instances. Streaming algorithms are another approach for dealing with large graphs. In this model, the graph is presented one-by-one in a stream of edge insertions or deletions and the algorithms are permitted to use only a limited amount of memory. Often, the solution to an optimization problem on graphs can require up to a linear amount of space with respect to the number of nodes, resulting in a trivial lower bound for the space requirement of any streaming algorithm for those problems. Computing a matching, i. e., a subset of edges where no two edges are incident to a common node, is an example which has recently attracted a lot of attention in the streaming setting. If we are interested in the size (or weight in case of weighted graphs) of a matching, it is possible to break this linear bound. We focus on so-called dynamic graph streams where edges can be inserted and deleted and reduce the problem of estimating the weight of a matching to the problem of estimating the size of a maximum matching with a small loss in the approximation factor. In addition, we present the first dynamic graph stream algorithm for estimating the size of a matching in graphs which are locally sparse. On the negative side, we prove a space lower bound of streaming algorithms that estimate the size of a maximum matching with a small approximation factor

Algorithms for the Maximum Independent Set Problem

This thesis focuses mainly on the Maximum Independent Set (MIS) problem. Some related graph theoretical combinatorial problems are also considered. As these problems are generally NP-hard, we study their complexity in hereditary graph classes, i.e. graph classes defined by a set F of forbidden induced subgraphs. We revise the literature about the issue, for example complexity results, applications, and techniques tackling the problem. Through considering some general approach, we exhibit several cases where the problem admits a polynomial-time solution. More specifically, we present polynomial-time algorithms for the MIS problem in: + some subclasses of $S_{2;j;k}$-free graphs (thus generalizing the classical result for $S_{1;2;k}$-free graphs); + some subclasses of $tree_{k}$-free graphs (thus generalizing the classical results for subclasses of P5-free graphs); + some subclasses of $P_{7}$-free graphs and $S_{2;2;2}$-free graphs; and various subclasses of graphs of bounded maximum degree, for example subcubic graphs. Our algorithms are based on various approaches. In particular, we characterize augmenting graphs in a subclass of $S_{2;k;k}$-free graphs and a subclass of $S_{2;2;5}$-free graphs. These characterizations are partly based on extensions of the concept of redundant set [125]. We also propose methods finding augmenting chains, an extension of the method in [99], and finding augmenting trees, an extension of the methods in [125]. We apply the augmenting vertex technique, originally used for $P_{5}$-free graphs or banner-free graphs, for some more general graph classes. We consider a general graph theoretical combinatorial problem, the so-called Maximum -Set problem. Two special cases of this problem, the so-called Maximum F-(Strongly) Independent Subgraph and Maximum F-Induced Subgraph, where F is a connected graph set, are considered. The complexity of the Maximum F-(Strongly) Independent Subgraph problem is revised and the NP-hardness of the Maximum F-Induced Subgraph problem is proved. We also extend the augmenting approach to apply it for the general Maximum Π -Set problem. We revise on classical graph transformations and give two unified views based on pseudo-boolean functions and αff-redundant vertex. We also make extensive uses of α-redundant vertices, originally mainly used for $P_{5}$-free graphs, to give polynomial solutions for some subclasses of $S_{2;2;2}$-free graphs and $tree_{k}$-free graphs. We consider some classical sequential greedy heuristic methods. We also combine classical algorithms with αff-redundant vertices to have new strategies of choosing the next vertex in greedy methods. Some aspects of the algorithms, for example forbidden induced subgraph sets and worst case results, are also considered. Finally, we restrict our attention on graphs of bounded maximum degree and subcubic graphs. Then by using some techniques, for example ff-redundant vertex, clique separator, and arguments based on distance, we general these results for some subclasses of $S_{i;j;k}$-free subcubic graphs

Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions