NP-hard problems in hierarchical-tree clustering
Acta Informatica
Cluster analysis and mathematical programming
Mathematical Programming: Series A and B - Special issue: papers from ismp97, the 16th international symposium on mathematical programming, Lausanne EPFL
Approximating dissimilarities by quasi-ultrametrics
Proceedings of the conference on Discrete metric spaces
The k-weak hierarchical representations: an extension of the indexed closed weak hierarchies
Discrete Applied Mathematics - Special issue: The 1998 conference on ordinal and symbolic data analysis (OSDA '98)
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Sub-dominant theory provides efficient tools for clustering. However, it classically works only for ultrametrics and ad hoc extensions like Jardine and Sibson's 2-ultrametrics. In this paper we study the extension of the notion of sub-dominant to other distance models in classification accounting for overlapping clusters. We prove that a given dissimilarity admits one and only one lower-maximal quasi-ultrametric and one and only one lower-maximal weak k-ultrametric. In addition, we also prove the existence of (several) lower-maximal strongly Robinsonian dissimilarities. The construction of the lower-maximal weak k-ultrametric (for k=2) and quasi-ultrametric can be performed in polynomial time.