This paper introduces the hypergraph transversal problem along with thefollowing iterative solutions: naive, branch and bound, and dynamic exponentialtime (NC-D). Odometers are introduced along with the functions that manipulatethem. The traditional definitions of hyperedge, hypergraph, etc., are redefined interms of odometers and lists. All algorithms and functions necessary to implementthe solution are presented along with techniques to validate and test the results.Lastly, parallelization advanced applications, and future research directions areexamined.

Ensembles of graphs arise naturally in many applications, for example, the temporal evolution of social contacts or computer communications, tissue-specific protein interaction networks, annual citation or co-authorship networks in a field, or a family of high-likelihood Bayesian networks inferred from systems biology data. Several techniques have been developed to analyze such ensembles. A canonical problem is that of computing communities that are persistent across the ensemble. This problem is usually formulated as one of computing dense subgraphs (communities) that are frequent, i.e., appear in many graphs in the ensemble. In this thesis, we seek to find "unstable communities" which are the antithesis of frequent, dense subgraphs. Informally, an unstable community is a set of nodes that induces highly-varying subgraphs in the ensemble. In other words, the graphs in the ensemble disagree about the precise pairwise connections among these nodes. The primary contribution of this dissertation is to introduce the concept of unstable communities as a novel problem in the field of graph mining. Specifically, it presents three approaches to mathematically formulate the concept of unstable communities, devises algorithms for computing such communities in a given ensemble of networks, and shows the usefulness of this concept in a variety of settings. Our first definition of unstable community relies on two parameters: the first ensures that a node set induces several different subgraphs in the ensemble and the second guarantees that each of these subgraphs occurs in a large number of graphs in the ensemble. We present two algorithms to enumerate unstable communities that match this definition. The first approach, ClustMiner, is a heuristic that transforms the problem into one of computing dense subgraphs in a single graph that summarizes the ensemble. The second approach, UCMiner, is guaranteed to enumerate all maximal unstable communities correctly. We apply both approaches to systems biology datasets to demonstrate that UCMiner is superior to ClustMiner in the sense that ClustMiner's output contains node sets that are not unstable while also missing several communities computed by UCMiner. We find several node sets that capture the uncertain connectivity of genes in relevant protein complexes, suggesting that further experiments may be required to precisely discern their interaction patterns. Our second and third definitions of unstable community rely on a novel concept of (scaled) subgraph divergence, a formulation that uses the concept of relative entropy to measure the instability of a community. We propose another algorithm, SDMiner, that can exactly enumerate all maximal unstable communities with small (scaled) subgraph divergence. We perform extensive experiments on social network datasets to show that we can discover UCs that capture the main structural variations of the given set of networks and also provide us with interesting and relevant insights about these datasets. / Ph. D.

The overall goal of the field of natural language processing is to facilitate the communication between humans and computers, and to help humans with natural language problems such as translation. In this thesis, we focus on semantic language processing. Modelling semantics – the meaning of natural language – requires both a structure to hold the semantic information and a device that can enforce rules on the structure to ensure well-formed semantics while not being too computationally heavy. The devices used in natural language processing are preferably weighted to allow for comparison of the alternative semantic interpretations outputted by a device. The structure employed here is the abstract meaning representation (AMR). We show that AMRs representing well-formed semantics can be generated while leaving out AMRs that are not semantically well-formed. For this purpose, we use a type of graph grammar called contextual hyperedge replacement grammar (CHRG). Moreover, we argue that a more well-known subclass of CHRG – the hyperedge replacement grammar (HRG) – is not powerful enough for AMR generation. This is due to the limitation of HRG when it comes to handling co-references, which in its turn depends on the fact that HRGs only generate graphs of bounded treewidth. Furthermore, we also address the N best problem, which is as follows: Given a weighted device, return the N best (here: smallest-weighted, or more intuitively, smallest-errored) structures. Our goal is to solve the N best problem for devices capable of expressing sophisticated forms of semantic representations such as CHRGs. Here, however, we merely take a first step consisting in developing methods for solving the N best problem for weighted tree automata and some types of weighted acyclic hypergraphs.

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This thesis presents work done on compressed graph representations via hyperedge replacement grammars. It comprises two main parts. Firstly the RePair compression scheme, known for strings and trees, is generalized to graphs using graph grammars. Given an object, the scheme produces a small context-free grammar generating the object (called a “straight-line grammar”). The theoretical foundations of this generalization are presented, followed by a description of a prototype implementation. This implementation is then evaluated on real-world and synthetic graphs. The experiments show that several graphs can be compressed stronger by the new method, than by current state-of-the-art approaches. The second part considers algorithmic questions of straight-line graph grammars. Two algorithms are presented to traverse the graph represented by such a grammar. Both algorithms have advantages and disadvantages: the first one works with any grammar but its runtime per traversal step is dependent on the input grammar. The second algorithm only needs constant time per traversal step, but works for a restricted class of grammars and requires quadratic preprocessing time and space. Finally speed-up algorithms are considered. These are algorithms that can decide specific problems in time depending only on the size of the compressed representation, and might thus be faster than a traditional algorithm would on the decompressed structure. The idea of such algorithms is to reuse computation already done for the rules of the grammar. The possible speed-ups achieved this way is proportional to the compression ratio of the grammar. The main results here are a method to answer “regular path queries”, and to decide whether two grammars generate isomorphic trees.

Access to and understanding data plays an essential role in the increasingly digital world. Representation and analysis of relations between various data entities, i.e., graph and network structures in the data, is an important problem for various industries. In contrast to simple graphs that focus on edges with two endpoints only, a hypergraph provides a natural method to represent multi-way interactions with an arbitrary number of endpoints for each edge, and it can be a better alternative than a bipartite graph for comparable applications. However, traditional approaches for visually representing hypergraphs are purely static diagrams without support for interaction, which can be difficult to perceive and do not scale well with regard to the number of nodes and edges. They are not adequate for the representation and interactive exploration of large or dense hypergraph data sets found in real-world applications. The ISOVIS (Information and Software Visualisation) research group at Linnaeus University has previously introduced a novel radial visualization approach for undirected hypergraphs called Onion. The Onion tool focuses on solving the issues of edge clutter, overlaps, and edge crossings. However, certain open challenges and suggestions for improvements were identified for the respective implementation, and there is an opportunity to fill a gap in the hypergraph visualization research by building upon the original Onion approach study. In this thesis project, we implement the new version of the Onion approach based on the principles and challenges established previously. The contributions of this work include evidence regarding the effectiveness and efficiency of a hypergraph comparison technique, the usability of edge bundling in the context of hypergraph exploration tasks, and the scalability of the interactive visualization through an entirely new web-based version of the Onion approach. To obtain the respective results, the new implementation is applied for two case studies involving real-world data sets, and further validated through a user study with several participants. The results of this work can be helpful for researchers of network visualization and practitioners in need of approaches for representing and exploring data that can be modeled as hypergraphs.

The formalized and algorithmic study of human language within the field of Natural Language Processing (NLP) has motivated much theoretical work in the related field of formal languages, in particular the subfields of grammar and automata theory. Motivated and informed by NLP, the papers in this thesis explore the connections between expressibility – that is, the ability for a formal system to define complex sets of objects – and algorithmic complexity – that is, the varying amount of effort required to analyse and utilise such systems. Our research studies formal systems working not just on strings, but on more complex structures such as trees and graphs, in particular syntax trees and semantic graphs. The field of mildly context-sensitive languages concerns attempts to find a useful class of formal languages between the context-free and context-sensitive. We study formalisms defining two candidates for this class; tree-adjoining languages and the languages defined by linear context-free rewriting systems. For the former, we specifically investigate the tree languages, and define a subclass and tree automaton with linear parsing complexity. For the latter, we use the framework of parameterized complexity theory to investigate more deeply the related parsing problems, as well as the connections between various formalisms defining the class. The field of semantic modelling aims towards formally and accurately modelling not only the syntax of natural language statements, but also the meaning. In particular, recent work in semantic graphs motivates our study of graph grammars and graph parsing. To the best of our knowledge, the formalism presented in Paper III of this thesis is the first graph grammar where the uniform parsing problem has polynomial parsing complexity, even for input graphs of unbounded node degree.