Combinatorial optimisation and hierarchical classifications

Abstract

This paper is devoted to some selected topics relating Combinatorial Optimization and Hierarchical Classification. It is oriented toward extensions of the standard classification schemes (the hierarchies): pyramids, quasi-hierarchies, circular clustering, rigid clustering and others. Bijection theorems between these models and dissimilarity models allow to state some clustering problems as optimization problems. Within the galaxy of optimization we have especially discussed the following: NP-completeness results and search for polynomial instances; problems solved in a polynomial time (e.g. subdominant theory); design, analysis and applications of algorithms. In contrast with the orientation to "new" clustering problems, the last part discusses some standard algorithmic approaches.

Ultrametric Watersheds

“…-Several generalisations of hierarchical clustering have been proposed in the literature [1]. An interesting direction of research is to see how to extend in the same way the topological watershed approach, for example for allowing regions to overlap.…”

“…One of the fundamental theorems for hierarchical clustering states that there exists a one-to-one correspondance between the set of indexed hierarchical classification and a particular subset of dissimilarity measures called ultrametric distances; This theorem is generally attributed to Johnson [4], Jardine et al [5] and Benzécri [3]. Since then, numerous generalisations of that bijection theorem have been proposed (see [1] for a recent review).…”

“…Following [1], we can say that, independently of its theoretical interest, such a bijection theorem is useful in practice. Any hierarchical segmentation problem is a priori heterogeneous: assign to an edge-weighted graph an indexed hierarchy.…”

Topographic spaces over ordered monoids

“…Now we recall the definitions of generalized metric and related concepts [3,5,14,15,18]. Definition 2.1.…”

Clustering of high throughput gene expression data

“…They are used when it is impossible or too time costly to find an optimal solution to a problem. Metaheuristics are generally used in partition based clustering and are rarely used in HC (Barthelemy et al, 2007). Genetic algorithms (GA), ant colony optimization (ACO), Tabu Search (TS), and simulated annealing (SA) are some widely used metaheuristics.…”