32 research outputs found
Efficient parameterized algorithms on structured graphs
In der klassischen Komplexitätstheorie werden worst-case Laufzeiten von Algorithmen typischerweise einzig abhängig von der Eingabegröße angegeben. In dem Kontext der parametrisierten Komplexitätstheorie versucht man die Analyse der Laufzeit dahingehend zu verfeinern, dass man zusätzlich zu der Eingabengröße noch einen Parameter berücksichtigt, welcher angibt, wie strukturiert die Eingabe bezüglich einer gewissen Eigenschaft ist. Ein parametrisierter Algorithmus nutzt dann diese beschriebene Struktur aus und erreicht so eine Laufzeit, welche schneller ist als die eines besten unparametrisierten Algorithmus, falls der Parameter klein ist.
Der erste Hauptteil dieser Arbeit führt die Forschung in diese Richtung weiter aus und untersucht den Einfluss von verschieden Parametern auf die Laufzeit von bekannten effizient lösbaren Problemen. Einige vorgestellte Algorithmen sind dabei adaptive Algorithmen, was bedeutet, dass die Laufzeit von diesen Algorithmen mit der Laufzeit des besten unparametrisierten Algorithm für den größtmöglichen Parameterwert übereinstimmt und damit theoretisch niemals schlechter als die besten unparametrisierten Algorithmen und übertreffen diese bereits für leicht nichttriviale Parameterwerte.
Motiviert durch den allgemeinen Erfolg und der Vielzahl solcher parametrisierten Algorithmen, welche eine vielzahl verschiedener Strukturen ausnutzen, untersuchen wir im zweiten Hauptteil dieser Arbeit, wie man solche unterschiedliche homogene Strukturen zu mehr heterogenen Strukturen vereinen kann. Ausgehend von algebraischen Ausdrücken, welche benutzt werden können, um von Parametern beschriebene Strukturen zu definieren, charakterisieren wir klar und robust heterogene Strukturen und zeigen exemplarisch, wie sich die Parameter tree-depth und modular-width heterogen verbinden lassen. Wir beschreiben dazu effiziente Algorithmen auf heterogenen Strukturen mit Laufzeiten, welche im Spezialfall mit den homogenen Algorithmen übereinstimmen.In classical complexity theory, the worst-case running times of algorithms depend solely on the size of the input. In parameterized complexity the goal is to refine the analysis of the running time of an algorithm by additionally considering a parameter that measures some kind of structure in the input. A parameterized algorithm then utilizes the structure described by the parameter and achieves a running time that is faster than the best general (unparameterized) algorithm for instances of low parameter value.
In the first part of this thesis, we carry forward in this direction and investigate the influence of several parameters on the running times of well-known tractable problems.
Several presented algorithms are adaptive algorithms, meaning that they match the running time of a best unparameterized algorithm for worst-case parameter values. Thus, an adaptive parameterized algorithm is asymptotically never worse than the best unparameterized algorithm, while it outperforms the best general algorithm already for slightly non-trivial parameter values.
As illustrated in the first part of this thesis, for many problems there exist efficient parameterized algorithms regarding multiple parameters, each describing a different kind of structure.
In the second part of this thesis, we explore how to combine such homogeneous structures to more general and heterogeneous structures.
Using algebraic expressions, we define new combined graph classes
of heterogeneous structure in a clean and robust way, and we showcase this for the heterogeneous merge of the parameters tree-depth and modular-width, by presenting parameterized algorithms
on such heterogeneous graph classes and getting running times that match the homogeneous cases throughout
Hypergraphs with Polynomial Representation: Introducing -splits
Inspired by the split decomposition of graphs and rank-width, we introduce
the notion of -splits. We focus on the family of -splits of a graph of
order , and we prove that it forms a hypergraph with several properties. We
prove that such hypergraphs can be represented using only
of its hyperedges, despite its potentially exponential number of hyperedges. We
also prove that there exist hypergraphs that need at least
hyperedges to be represented, using a generalization of set orthogonality
Generalizations of comparability graphs
2022 Summer.Includes bibliographical references.In rational decision-making models, transitivity of preferences is an important principle. In a transitive preference, one who prefers x to y and y to z must prefer x to z. Many preference relations, including total order, weak order, partial order, and semiorder, are transitive. As a preference which is transitive yet not all pairs of elements are comparable, partial orders have been studied extensively. In graph theory, a comparability graph is an undirected graph which connects all comparable elements in a partial order. A transitive orientation is an assignment of direction to every edge so that the resulting directed graph is transitive. A graph is transitive if there is such an assignment. Comparability graphs are a class of graphs where clique, coloring, and many other optimization problems are solved by polynomial algorithms. It also has close connections with other classes of graphs, such as interval graphs, permutation graphs, and perfect graphs. In this dissertation, we define new measures for transitivity to generalize comparability graphs. We introduce the concept of double threshold digraphs together with a parameter λ which we define as our degree of transitivity. We also define another measure of transitivity, β, as the longest directed path such that there is no edge from the first vertex to the last vertex. We present approximation algorithms and parameterized algorithms for optimization problems and demonstrate that they are efficient for "almost-transitive" preferences
A framework for structuring prerequisite relations between concepts in educational textbooks
In our age we are experiencing an increasing availability of digital educational resources and self-regulated learning. In this scenario, the development of automatic strategies for organizing the knowledge embodied in educational resources has a tremendous potential for building personalized learning paths and applications such as intelligent textbooks and recommender systems of learning materials. To this aim, a straightforward approach consists in enriching the educational materials with a concept graph, i.a. a knowledge structure where key concepts of the subject matter are represented as nodes and prerequisite dependencies among such concepts are also explicitly represented. This thesis focuses therefore on prerequisite relations in textbooks and it has two main research goals. The first goal is to define a methodology for systematically annotating prerequisite relations in textbooks, which is functional for analysing the prerequisite phenomenon and for evaluating and training automatic methods of extraction. The second goal concerns the automatic extraction of prerequisite relations from textbooks. These two research goals will guide towards the design of PRET, i.e. a comprehensive framework for supporting researchers involved in this research issue. The framework described in the present thesis allows indeed researchers to conduct the following tasks: 1) manual annotation of educational texts, in order to create datasets to be used for machine learning algorithms or for evaluation as gold standards; 2) annotation analysis, for investigating inter-annotator agreement, graph metrics and in-context linguistic features; 3) data visualization, for visually exploring datasets and gaining insights of the problem that may lead to improve algorithms; 4) automatic extraction of prerequisite relations. As for the automatic extraction, we developed a method that is based on burst analysis of concepts in the textbook and we used the gold dataset with PR annotation for its evaluation, comparing the method with other metrics for PR extraction
Separability and Vertex Ordering of Graphs
Many graph optimization problems, such as finding an optimal coloring, or a largest clique, can be solved by a divide-and-conquer approach. One such well-known technique is decomposition by clique separators where a graph is decomposed into special induced subgraphs along their clique separators. While the most common practice of this method employs minimal clique separators, in this work we study other variations as well. We strive to characterize their structure and in particular the bound on the number of atoms. In fact, we strengthen the known bounds for the general clique cutset decomposition and the minimal clique separator decomposition. Graph ordering is the arrangement of a graph’s vertices according to a certain logic and is a useful tool in optimization problems. Special types of vertices are often recognized in graph classes, for instance it is well-known every chordal graph contains a simplicial vertex. Vertex-ordering, based on such properties, have originated many linear time algorithms. We propose to define a new family named SE-Class such that every graph belonging to this family inherently contains a simplicial extreme, that is a vertex which is either simplicial or has exactly two neighbors which are non-adjacent. Our family lends itself to an ordering based on simplicial extreme vertices (named SEO) which we demonstrate to be advantageous for the coloring and maximum clique problems. In addition, we examine the relation of SE-Class to the family of (Even-Hole, Kite)-free graphs and show a linear time generation of SEO for (Even-Hole, Diamond, Claw)-free graphs. We showcase the applications of those two core tools, namely clique-based decomposition and vertex ordering, on the (Even-Hole, Kite)-free family
3-uniform hypergraphs: modular decomposition and realization by tournaments
Let be a 3-uniform hypergraph. A tournament defined on is
a realization of if the edges of are exactly the 3-element subsets of
that induce 3-cycles. We characterize the 3-uniform hypergraphs that
admit realizations by using a suitable modular decomposition
Formal Linguistic Models and Knowledge Processing. A Structuralist Approach to Rule-Based Ontology Learning and Population
2013 - 2014The main aim of this research is to propose a structuralist approach for knowledge processing by means of ontology learning and population, achieved starting from unstructured and structured texts. The method suggested includes distributional semantic approaches and NL formalization theories, in order to develop a framework, which relies upon deep linguistic analysis... [edited by author]XIII n.s
Unavoidable induced subgraphs in large graphs with no homogeneous sets
A homogeneous set of an -vertex graph is a set of vertices () such that every vertex not in is either complete or
anticomplete to . A graph is called prime if it has no homogeneous set. A
chain of length is a sequence of vertices such that for every vertex
in the sequence except the first one, its immediate predecessor is its unique
neighbor or its unique non-neighbor among all of its predecessors. We prove
that for all , there exists such that every prime graph with at least
vertices contains one of the following graphs or their complements as an
induced subgraph: (1) the graph obtained from by subdividing every
edge once, (2) the line graph of , (3) the line graph of the graph in
(1), (4) the half-graph of height , (5) a prime graph induced by a chain of
length , (6) two particular graphs obtained from the half-graph of height
by making one side a clique and adding one vertex.Comment: 13 pages, 3 figure
Social and Semantic Contexts in Tourist Mobile Applications
The ongoing growth of the World Wide Web along with the increase possibility of access information through a variety of devices in mobility, has defi nitely changed the way users acquire, create, and personalize information, pushing innovative strategies for annotating and organizing it.
In this scenario, Social Annotation Systems have quickly gained a huge popularity, introducing millions of metadata on di fferent Web resources following a bottom-up approach, generating free and democratic mechanisms of classi cation, namely folksonomies. Moving away from hierarchical classi cation schemas, folksonomies represent also a meaningful mean for identifying similarities among users, resources and tags. At any rate, they suff er from several limitations, such as the lack of specialized tools devoted to manage, modify, customize and visualize them as well as the lack of an explicit semantic, making di fficult for users to bene fit from them eff ectively. Despite appealing promises of Semantic Web technologies, which were intended to explicitly formalize the knowledge within a particular domain in a top-down manner, in order to perform intelligent integration and reasoning on it, they are still far from reach their objectives, due to di fficulties in knowledge acquisition and annotation bottleneck.
The main contribution of this dissertation consists in modeling a novel conceptual framework that exploits both social and semantic contextual dimensions, focusing on the domain of tourism and cultural heritage. The primary aim of our assessment is to evaluate the overall user satisfaction and the perceived quality in use thanks to two concrete case studies. Firstly, we concentrate our attention on contextual information and navigation, and on authoring tool; secondly, we provide a semantic mapping of tags of the system folksonomy, contrasted and compared to the expert users' classi cation, allowing a bridge between social and semantic knowledge according to its constantly mutual growth.
The performed user evaluations analyses results are promising, reporting a high level of agreement on the perceived quality in use of both the applications and of the speci c analyzed features, demonstrating that a social-semantic contextual model improves the general users' satisfactio