List rankings and on-line list rankings of graphs

A $k$-ranking of a graph $G$ is a labeling of its vertices from $\{1,\ldots,k\}$ such that any nontrivial path whose endpoints have the same label contains a larger label. The least $k$ for which $G$ has a $k$-ranking is the ranking number of $G$, also known as tree-depth. The list ranking number of $G$ is the least $k$ such that if each vertex of $G$ is assigned a set of $k$ potential labels, then $G$ can be ranked by labeling each vertex with a label from its assigned list. Rankings model a certain parallel processing problem in manufacturing, while the list ranking version adds scheduling constraints. We compute the list ranking number of paths, cycles, and trees with many more leaves than internal vertices. Some of these results follow from stronger theorems we prove about on-line versions of list ranking, where each vertex starts with an empty list having some fixed capacity, and potential labels are presented one by one, at which time they are added to the lists of certain vertices; the decision of which of these vertices are actually to be ranked with that label must be made immediately.Comment: 16 pages, 3 figure

Learning preferences for large scale multi-label problems

Despite that the majority of machine learning approaches aim to solve binary classification problems, several real-world applications require specialized algorithms able to handle many different classes, as in the case of single-label multi-class and multi-label classification problems. The Label Ranking framework is a generalization of the above mentioned settings, which aims to map instances from the input space to a total order over the set of possible labels. However, generally these algorithms are more complex than binary ones, and their application on large-scale datasets could be untractable. The main contribution of this work is the proposal of a novel general online preference-based label ranking framework. The proposed framework is able to solve binary, multi-class, multi-label and ranking problems. A comparison with other baselines has been performed, showing effectiveness and efficiency in a real-world large-scale multi-label task

Label Ranking with Probabilistic Models

Diese Arbeit konzentriert sich auf eine spezielle Prognoseform, das sogenannte Label Ranking. Auf den Punkt gebracht, kann Label Ranking als eine Erweiterung des herkömmlichen Klassifizierungproblems betrachtet werden. Bei einer Anfrage (z. B. durch einen Kunden) und einem vordefinierten Set von Kandidaten Labels (zB AUDI, BMW, VW), wird ein einzelnes Label (zB BMW) zur Vorhersage in der Klassifizierung benötigt, während ein komplettes Ranking aller Label (zB BMW> VW> Audi) für das Label Ranking erforderlich ist. Da Vorhersagen dieser Art, bei vielen Problemen der realen Welt nützlich sind, können Label Ranking-Methoden in mehreren Anwendungen, darunter Information Retrieval, Kundenwunsch Lernen und E-Commerce eingesetzt werden. Die vorliegende Arbeit stellt eine Auswahl an Methoden für Label-Ranking vor, die Maschinelles Lernen mit statistischen Bewertungsmodellen kombiniert. Wir konzentrieren wir uns auf zwei statistische Ranking-Modelle, das Mallows- und das Plackett-Luce-Modell und zwei Techniken des maschinellen Lernens, das Beispielbasierte Lernen und das Verallgemeinernde Lineare Modell

Uniqueness and minimal obstructions for tree-depth

A k-ranking of a graph G is a labeling of the vertices of G with values from {1,...,k} such that any path joining two vertices with the same label contains a vertex having a higher label. The tree-depth of G is the smallest value of k for which a k-ranking of G exists. The graph G is k-critical if it has tree-depth k and every proper minor of G has smaller tree-depth. We establish partial results in support of two conjectures about the order and maximum degree of k-critical graphs. As part of these results, we define a graph G to be 1-unique if for every vertex v in G, there exists an optimal ranking of G in which v is the unique vertex with label 1. We show that several classes of k-critical graphs are 1-unique, and we conjecture that the property holds for all k-critical graphs. Generalizing a previously known construction for trees, we exhibit an inductive construction that uses 1-unique k-critical graphs to generate large classes of critical graphs having a given tree-depth.Comment: 14 pages, 4 figure