TASK DEFINITION: In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.
PROBLEM: In many combinatorial problems one may need to model the diversity or similarity of sets of assignments. For example, one may wish to maximise or minimise the number of distinct values in a solution. To formulate problems of this type we can use soft variants of the well known AllDifferent and AllEqual constraints. We present a taxonomy of six soft global constraints, generated by combining the two latter ones and the two standard cost functions, which are either maximised or minimised. We characterise the complexity of achieving arc and bounds consistency on these constraints, resolving those cases for which NP-hardness was neither proven nor disproven. In particular, we explore in depth the constraint ensuring that at least k pairs of variables have a common value. We show that achieving arc consistency is NP-hard, however bounds consistency can be achieved in polynomial time through dynamic programming. Moreover, we show that the maximum number of pairs of equal variables can be approximated by a factor of 12 with a linear time greedy algorithm. Finally, we provide a fixed parameter tractable algorithm with respect to the number of values appearing in more than two distinct domains. Interestingly, this taxonomy shows that enforcing equality is harder than enforcing difference.

SOLUTION: Soft Constraints of Difference and Equality

PROBLEM: Nowadays ontologies present a growing interest in Data Fusion applications. As a matter of fact, the ontologies are seen as a semantic tool for describing and reasoning about sensor data, objects, relations and general domain theories. In addition, uncertainty is perhaps one of the most important characteristics of the data and information handled by Data Fusion. However, the fundamental nature of ontologies implies that ontologies describe only asserted and veracious facts of the world. Different probabilistic, fuzzy and evidential approaches already exist to fill this gap; this paper recaps the most popular tools. However none of the tools meets exactly our purposes. Therefore, we constructed a Dempster-Shafer ontology that can be imported into any specific domain ontology and that enables us to instantiate it in an uncertain manner. We also developed a Java application that enables reasoning about these uncertain ontological instances.

SOLUTION: Uncertainty in Ontologies: Dempster-Shafer Theory for Data Fusion Applications

PROBLEM: We develop a worst-case analysis of aggregation of binary classifier ensembles in a transductive setting, for a broad class of losses including but not limited to all convex surrogates. The result is a family of parameter-free ensemble aggregation algorithms, which are as efficient as linear learning and prediction for convex risk minimization but work without any relaxations whatsoever on many nonconvex losses like the 0-1 loss. The prediction algorithms take a familiar form, applying "link functions" to a generalized notion of ensemble margin, but without the assumptions typically made in margin-based learning all this structure follows from a minimax interpretation of loss minimization.

SOLUTION:
Minimax Binary Classifier Aggregation with General Losses