NP-hard approximation problems in overlapping clustering

In this paper we prove that the approximation of a dissimilarity by an indexed pseudo-hierarchy (also called a pyramid) or an indexed quasi-hierarchy (also called an indexed weak hierarchy) is an NP-hard problem for any Lp-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.