Jean-Pierre Barthélemy

Human centered processes and decision support systems

Abstract This paper emphasizes the role of human factors in Decision Support Systems and related assisting tools that can be used in the Operational Research field. It links both historical information and real life realizations concerning the human centered processes. The historical points mentioned in the paper give only partial emphasis, according to the feeling of the authors. The aim, here, is essentially to review some tools (e.g., utility theory, cognitive modeling, etc.) that are or might be used to tackle new problems in the context of anthropocentered systems, especially when considering the recent evolution of Information Systems towards distributed ones. Several real-life problems (mostly in an industrial setting) are reviewed. They all concern applications on which the authors have worked (or are working) together.

From copair hypergraphs to median graphs with latent vertices

Abstract The purpose of this paper is to extend the Buneman construction of partially labelled trees to the general case. This extension is related with the characterization of median graphs by Mulder and Schrijver. In the first section, we construct a graph G ( H ) associated with a copair hypergraph H on a finite set X and define the notion of a median graph with latent vertices (called X -median graph). The latent vertices (i.e. the vertices who are not labelled by elements of X ) are obtained by iterating the median operation from actual (labelled) vertices. In the second section, we prove that the graph G ( H ) is an X -median graph. Then, in the last section, we study some special cases, the Buneman result is reobtained and the hypergraphs whose associated graphs are Hasse diagrams of distributive lattices are characterized.

NP-hard Approximation Problems in Overlapping Clustering

L p-norm (p < ∞). These problems also correspond to the approximation by a strongly Robinson dissimilarity or by a dissimilarity fulfilling the four-point inequality (Bandelt 1992; Diatta and Fichet 1994). The results are extended to circular strongly Robinson dissimilarities, indexed k-hierarchies (Jardine and Sibson 1971, pp. 65-71), and to proper dissimilarities satisfying the Bertrand and Janowitz (k + 2)-point inequality (Bertrand and Janowitz 1999). Unidimensional scaling (linear or circular) is reinterpreted as a clustering problem and its hardness is established, but only for the L1 norm.

Dictatorial consensus functions on n-trees

Abstract The original ‘independence of irrelevant alternatives’ axiom of K. Arrow has a natural analog when translated from the classical weak order (preference relation) case to n-trees. Using this translated independence axiom for n-trees, it is surprising that Arrows Impossibility Theorem does not follow. Specifically, there exist consensus functions for n-trees that satisfy the independence and Pareto conditions but are not dictatorships. Conversely, a dictatorship must clearly satisfy the Pareto condition but not necessarily independence. In this note it is shown that a consensus function for n-trees is a dictatorship and satisfies independence if and only if it is a projection function.

Arrow's theorem: unusual domains and extended codomains

Abstract In this paper we establish Arrows theorem in a general ordinal case. When some configurations are allowed in the domain and if this domain is included in the codomain, the only social functions satisfying the independence condition and the weak Pareto Principle are the absolute dictatorships or the absolute oligarchies.

Median graphs, parallelism and posets

A notion of parallelism is defined in finite median graphs and a number of properties about geodesics and the existence of cubes are obtained. Introducing sites as a double structure of partial order and graph on a set, it is shown that all median graphs can be constructed from sites and, in fact, that the categories of sites and pointed median graphs are equivalent, generalizing Birkhoffs duality.

Social Welfare and Aggregation Procedures: Combinatorial and Algorithmic Aspects

In this paper, we review some aspects about aggregation procedures. First some examples are given: Borda count, Condorcet rule, decisive procedures, Kemeny’s medians, Dogson procedure. Then a general definition of an aggregation procedure is proposed and a hierarchy of results (possible/impossible, computable/non computable, easy/hard) is illustrated by several examples. The last part of this paper is devoted to a formal theory of medians and a new possibility result is obtained for social welfare functions.

Binary clustering

In many clustering systems (hierarchies, pyramids and more generally weak hierarchies) clusters are generated by two elements only. This paper is devoted to such clustering systems (called binary clustering systems). It provides some basic properties, links with (closed) weak hierarchies and some qualitative versions of bijection theorems that occur in Numerical Taxonomy. Moreover, a way to associate a binary clustering system to every clustering system is discussed. Finally, introducing the notion of weak ultrametrics, a bijection between indexed weak hierarchies and weak ultrametrics is obtained (the standard theorem involves closed weak hierarchies and quasi-ultrametrics).

Average Consensus in Numerical Taxonomy and Some Generalizations

This paper is devoted to the notion of average consensus together with some generalizations involving L p -norms.

Graphs, L1-Metrics and Clustering

This paper is devoted to p-distances (i.e. proper dissimilarities that take only the values 0, p − 1, and p) in connection with non-hierarchical models in classification. Three kinds of topics are examined: the study of the p-distances that are embeddable into L 1-space; the characterization of several non-hierarchical clusterng models arising from particular p-distances (partitioning, quasi-partitioning and Robinsonian graphs) and finally, the proof of the NP-hardness of the approximation of a proper dissimilarity by these models.