6 research outputs found
NP-hardness of the computation of a median equivalence relation in classification (RĂ©gnier problem)
Ătant donnĂ©e une collection de relations dâĂ©quivalence (ou partitions), le problĂšme de RĂ©gnier consiste Ă dĂ©terminer une relation dâĂ©quivalence qui minimise lâĂ©loignement par rapport Ă . LâĂ©loignement est fondĂ© sur la distance de la diffĂ©rence symĂ©trique et mesure le nombre de dĂ©saccords entre ï et la relation dâĂ©quivalence considĂ©rĂ©e. Une telle relation dâĂ©quivalence minimisant lâĂ©loignement est appelĂ©e une relation dâĂ©quivalence mĂ©diane de . On montre ici la NP-difficultĂ© du problĂšme de RĂ©gnier, câest-Ă -dire du calcul dâune relation dâĂ©quivalence mĂ©diane dâune collection de relations dâĂ©quivalence, du moins quand le nombre de relations dâĂ©quivalence de est suffisamment grand.Given a collection of equivalence relations (or partitions), RĂ©gnierâs problem consists in computing an equivalence relation which minimizes the remoteness from . The remoteness is based on the symmetric difference distance and measures the number of disagreements between and the considered equivalence relation. Such an equivalence relation minimizing the remoteness is called a median equivalence relation of . We prove the NP-hardness of RĂ©gnierâs problem, i.e. the computation of a median equivalence relation of a collection of equivalence relations, at least when the number of equivalence relations of is large enough
Consensus theories: an oriented survey
This article surveys seven directions of consensus theories: Arrowian results, federation consensus rules, metric consensus rules, tournament solutions, restricted domains, abstract consensus theories, algorithmic and complexity issues. This survey is oriented in the sense that it is mainly â but not exclusively â concentrated on the most significant results obtained, sometimes with other searchers, by a team of French searchers who are or were full or associate members of the Centre d'Analyse et de MathĂ©matique Sociale (CAMS).Consensus theories ; Arrowian results ; aggregation rules ; metric consensus rules ; median ; tournament solutions ; restricted domains ; lower valuations ; median semilattice ; complexity
Fair Correlation Clustering in Forests
The study of algorithmic fairness received growing attention recently. This stems from the awareness that bias in the input data for machine learning systems may result in discriminatory outputs. For clustering tasks, one of the most central notions of fairness is the formalization by Chierichetti, Kumar, Lattanzi, and Vassilvitskii [NeurIPS 2017]. A clustering is said to be fair, if each cluster has the same distribution of manifestations of a sensitive attribute as the whole input set. This is motivated by various applications where the objects to be clustered have sensitive attributes that should not be over- or underrepresented. Most research on this version of fair clustering has focused on centriod-based objectives.
In contrast, we discuss the applicability of this fairness notion to Correlation Clustering. The existing literature on the resulting Fair Correlation Clustering problem either presents approximation algorithms with poor approximation guarantees or severely limits the possible distributions of the sensitive attribute (often only two manifestations with a 1:1 ratio are considered). Our goal is to understand if there is hope for better results in between these two extremes. To this end, we consider restricted graph classes which allow us to characterize the distributions of sensitive attributes for which this form of fairness is tractable from a complexity point of view.
While existing work on Fair Correlation Clustering gives approximation algorithms, we focus on exact solutions and investigate whether there are efficiently solvable instances. The unfair version of Correlation Clustering is trivial on forests, but adding fairness creates a surprisingly rich picture of complexities. We give an overview of the distributions and types of forests where Fair Correlation Clustering turns from tractable to intractable.
As the most surprising insight, we consider the fact that the cause of the hardness of Fair Correlation Clustering is not the strictness of the fairness condition. We lift most of our results to also hold for the relaxed version of the fairness condition. Instead, the source of hardness seems to be the distribution of the sensitive attribute. On the positive side, we identify some reasonable distributions that are indeed tractable. While this tractability is only shown for forests, it may open an avenue to design reasonable approximations for larger graph classes
Consensus theories: an oriented survey
URL des Documents de travail : http://ces.univ-paris1.fr/cesdp/cesdp2010.htmlDocuments de travail du Centre d'Economie de la Sorbonne 2010.57 - ISSN : 1955-611XThis article surveys seven directions of consensus theories: Arrowian results, federation consensus rules, metric consensus rules, tournament solutions, restricted domains, abstract consensus theories, algorithmic and complexity issues. This survey is oriented in the sense that it is mainly â but not exclusively â concentrated on the most significant results obtained, sometimes with other searchers, by a team of French searchers who are or were full or associate members of the Centre d'Analyse et de MathĂ©matique Sociale (CAMS).Cet article prĂ©sente une vue d'ensemble de sept directions de recherche en thĂ©orie du consensus : rĂ©sultats arrowiens, rĂšgles d'agrĂ©gation dĂ©finies au moyen de fĂ©dĂ©rations, rĂšgles dĂ©finies au moyen de distances, solutions de tournoi, domaines restreints, thĂ©ories abstraites du consensus, questions de complexitĂ© et d'algorithmique. Ce panorama est orientĂ© dans la mesure oĂč il prĂ©sente principalement â mais non exclusivement â les travaux les plus significatifs obtenus â quelquefois avec d'autres chercheurs â par une Ă©quipe de chercheurs français qui sont â ou ont Ă©tĂ© â membres plĂ©niers ou associĂ©s du Centre d'Analyse et de MathĂ©matique Sociale (CAMS)