Bruno Leclerc

Efficient and binary consensus functions on transitively valued relations

Abstract We give the general form of consensus functions on valued (or fuzzy) quasi-orders which satisfy two simple Arrow-like conditions of efficiency and binariness. Various previously known consensus results on the aggregation of preferences and mathematical classification may be derived from our result: directly, on quasi-orders (Mirkin), equivalences (Mirkin) or ultrametrics (Barthelemy, Leclerc and Monjardet); with further considerations, on complete quasi-orders or weak orders (Arrow; Mas-Collel and Sonnenschein) and orders (Brown).

Aggregation of fuzzy preferences: A theoretic Arrow-like approach

Starting from the recent theoretical study of Monjardet on Arrow-like characterizations of aggregation functions in finite lattice structures, we study the case of lattices of fuzzy objects (preferences, classifications,..). Here fuzzy preferences are defined in a fairly known way. Given a finite lattice P of preferences, the set J of its join-irreducible elements and a lattice C of values, a fuzzy preference is a mapping q:J →C satisfying the following compatibility condition: (∀j ∈J, I⊆J) j ⩽ V I ⇒ q(j)⩾ ∧ {q(i):i ∈ I}. It is equivalent to considered the Galois connections between P and C, or the residuated mappings from P into thdual C∗ of C. We particulaze Monjardets results to this case by studying the properties of the lattice Q when those of P and C are known. In a quite frequent case, we obtain specific results, including those of a 1984 paper on valued preorders.

Minimum spanning trees for tree metrics: abridgements and adjustments

Two properties of tree metrics are already known in the literature: tree metrics on a setX withn elements have 2n−3 degrees of freedom; a tree metric has Robinson form with regard to its minimum spanning tree (MST), or to any such MST if several of them exist. Starting from these results, we prove that a tree metrict is entirely defined by its restriction to some setB of 2n−3 entries. This set is easily determined from the table oft and includes then−1 entries of an MST. A fast method for the adjustment of a tree metric to any given metricd is then obtained. This method extends to dissimilarities.RésuméOn part de deux propriétés connues sur les distances arborées: leur nombre de degrés de liberté est 2n−3 lorsqu’elles sont définies sur un ensembleX àn éléments; une distance arborée a une propriété de type Robinsonien par rapport à son arbre minimum, ou à tout arbre minimum si elle en a plusieurs. Nous montrons alors qu’une distance arboréet est entèrement définie par sa restriction à un ensembleB de 2n−3 de ses entrées. Cet ensemble est facile à construire à partir de la table det et inclut un arbre minimum. Une méthode rapide pour l’ajustement d’une distance arborée à toute distanced donnée en est déduite; elle s’applique aussi aux dissimilarités.

Comparison of additive trees using circular orders.

It has been postulated that existing species have been linked in the past in a way that can be described using an additive tree structure. Any such tree structure reflecting species relationships is associated with a matrix of distances between the species considered which is called a distance matrix or a tree metric matrix. A circular order of elements of X corresponds to a circular (clockwise) scanning of the subset X of vertices of a tree drawn on a plane. This paper describes an optimal algorithm using circular orders to compare the topology of two trees given by their distance matrices. This algorithm allows us to compute the Robinson and Foulds topologic distance between two trees. It employs circular order tree reconstruction to compute an ordered bipartition table of the tree edges for both given distance matrices. These bipartition tables are then compared to determine the Robinson and Foulds topologic distance, known to be an important criterion of tree similarity. The described algorithm has optimal time complexity, requiring O(n(2)) time when performed on two n x n distance matrices. It can be generalized to get another optimal algorithm, which enables the strict consensus tree of k unrooted trees, given their distance matrices, to be constructed in O(kn(2)) time.

On some relations between 2-trees and tree metrics

Abstract A tree function (TF) t on a finite set X is a real function on the set of the pairs of elements of X satisfying the four-point condition: for all distinct x, y, z, w ∈ X, t(xy)+t(zw)⩽ max{t(xz) + t(yw), t(xw) + t(yz)}. Equivalently, t is representable by the lengths of the paths between the leaves of a valued tree Tl. TFs are a straightforward generalization of the tree dissimilarities and tree metrics of the literature. A graph Θ is a 2-tree if it belongs to the following class Q : an edge-graph belongs to Q : if Θ′ ∈ Q and yz is an edge of Θ′, then the graph obtained by the addition to Θl of a new vertex x adjacent to y and z belongs to Q . These graphs, and the more general k-trees, have been studied in the literature as generalizations of trees. It is first explicited here how to make a TF tΘ, d correspond to any positively valued 2-tree Θd on X. Then, given a tree dissimilarity t, the set Q(t) of the 2-trees Θ such that t = tΘ, t is studied. Any element of Q(t) gives a way of summarizing t by its restriction to a minimal subset of entries. Several characterizations and properties of the elements of Q(t) are given. We describe five classes of such elements, including two new ones. Associated with a dissimilarity of the general type, these classes of 2-trees lead to methods for the recognition and fitting of tree dissimilarities.

Medians for weight metrics in the covering graphs of semilattices

Abstract We consider the undirected covering graph G of a finite (meet) semilattice X endowed with a lower valuation. More precisely, our main concerns are the lower valuations associated to a weighting of the join-irreducible elements of X and the corresponding minimum path length metrics in G, which are frequently considered in the literature. Some results on the medians for such metrics are obtained, in relation with a lattice majority rule. Especially, these medians are characterized in the case where X is distributive. The unanimity, or Pareto, property is also investigated for such medians.

The median procedure in the semilattice of orders

Let X be a finite set; we are concerned with the problem of finding a consensus order P that summarizes an m-tuple (profile) P* of (partial) orders on X. A classical approach is to consider a distance function d on the set O of all the orders of X and to search to minimize the remoteness σ1 ≤i≤m d(P, Pi). We study some properties of this median procedure, and compare it with some other consensus approaches. Besides the classical symmetric difference metric, other distances are considered, and we particularly address the consequences for the consensus problem of the existence of a semilattice structure on the set O.

Consensus of classifications: the case of trees

We present a survey of the literature on the consensus of classification trees, based on a corpus (in progress) of about ninety papers.

Closure systems, implicational systems, overhanging relations and the case of hierarchical classification

Abstract Moore families and closure operators, especially those appearing in hierarchical classification, are considered here from the point of view of their related implicational systems and overhanging relations. The last ones, newly introduced here, generalize the “nesting relations” defined by Adams [J. Classification 3 (1986) 299] in the case of classification trees. Here we characterize overhanging relations by three axioms, and prove that they are in a one-to-one correspondence with Moore families. We study which properties of implicational systems and overhanging relations are general, and which are specific to hierarchical structures. We also characterize canonical implication bases of hierarchies and obtain a similar result for overhangings.

The residuation model for the ordinal construction of dissimilarities and other valued objects

The aim of this chapter is to present an ordinal model of valued objects, special cases of which appear in many contexts. The model lies on basic notions of ordered set theory: residuation or, equivalently, Galois connections; it is not new: explicitly proposed in fuzzy set theory by Achache (1982, 1988), it also underlies an order formalization of a Jardine and Sibson (1971) model given by Janowitz (1978; see also Barthelemy, Leclerc and Monjardet 1984a). Here, our main concern is to apply the model in order to obtain and study dissimilarities such as ultrametrics, Robinson or tree-compatible ones. Valued objects of other types, already considered in the literature, will be also given as examples: two types of valued non symmetric relations and two types of valued convex subsets. The chapter is neither a theoretical general presentation nor a detailed study of a few special cases. It is, tentatively, something between these extreme points of view. Some references are given to the reader interested to more details on a specific class of valued objects, or to more information about residuation (or Galois mappings) theory. In what follows, E will be a given finite set with n elements. Several families of combinatorial objects defined on E will be considered.