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 – but not exclusively – 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 Mathématique Sociale (CAMS).Consensus theories ; Arrowian results ; aggregation rules ; metric consensus rules ; median ; tournament solutions ; restricted domains ; lower valuations ; median semilattice ; complexity

Graph isomorphism completeness for trapezoid graphs

The complexity of the graph isomorphism problem for trapezoid graphs has been open over a decade. This paper shows that the problem is GI-complete. More precisely, we show that the graph isomorphism problem is GI-complete for comparability graphs of partially ordered sets with interval dimension 2 and height 3. In contrast, the problem is known to be solvable in polynomial time for comparability graphs of partially ordered sets with interval dimension at most 2 and height at most 2.Comment: 4 pages, 3 Postscript figure

Convexity in partial cubes: the hull number

We prove that the combinatorial optimization problem of determining the hull number of a partial cube is NP-complete. This makes partial cubes the minimal graph class for which NP-completeness of this problem is known and improves some earlier results in the literature. On the other hand we provide a polynomial-time algorithm to determine the hull number of planar partial cube quadrangulations. Instances of the hull number problem for partial cubes described include poset dimension and hitting sets for interiors of curves in the plane. To obtain the above results, we investigate convexity in partial cubes and characterize these graphs in terms of their lattice of convex subgraphs, improving a theorem of Handa. Furthermore we provide a topological representation theorem for planar partial cubes, generalizing a result of Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure

Dimension and cut vertices: an application of Ramsey theory

Motivated by quite recent research involving the relationship between the dimension of a poset and graph-theoretic properties of its cover graph, we show that for every $d\geq 1$, if $P$ is a poset and the dimension of a subposet $B$ of $P$ is at most $d$ whenever the cover graph of $B$ is a block of the cover graph of $P$, then the dimension of $P$ is at most $d+2$. We also construct examples which show that this inequality is best possible. We consider the proof of the upper bound to be fairly elegant and relatively compact. However, we know of no simple proof for the lower bound, and our argument requires a powerful tool known as the Product Ramsey Theorem. As a consequence, our constructions involve posets of enormous size.Comment: Final published version with updated reference

Characterization of image sets: the Galois Lattice approach

This paper presents a new method for supervised image classification. One or several landmarks are attached to each class, with the intention of characterizing it and discriminating it from the other classes. The different features, deduced from image primitives, and their relationships with the sets of images are structured and organized into a hierarchy thanks to an original method relying on a mathematical formalism called Galois (or Concept) Lattices. Such lattices allow us to select features as landmarks of specific classes. This paper details the feature selection process and illustrates this through a robotic example in a structured environment. The class of any image is the room from which the image is shot by the robot camera. In the discussion, we compare this approach with decision trees and we give some issues for future research

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 – but not exclusively – 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 Mathématique Sociale (CAMS).Cet article présente une vue d'ensemble de sept directions de recherche en théorie du consensus : résultats arrowiens, règles d'agrégation définies au moyen de fédérations, règles définies au moyen de distances, solutions de tournoi, domaines restreints, théories abstraites du consensus, questions de complexité et d'algorithmique. Ce panorama est orienté dans la mesure où il présente principalement – mais non exclusivement – les travaux les plus significatifs obtenus – quelquefois avec d'autres chercheurs – par une équipe de chercheurs français qui sont – ou ont été – membres pléniers ou associés du Centre d'Analyse et de Mathématique Sociale (CAMS)