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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.

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).

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