Summary
We consider a class of optimization problems of hierarchical-tree clustering and prove that these problems are NP-hard. The sequence of polynomial reductions and/or transformations used in our proof is based on relatively laborious graph-theoretical constructions and starts in the NP-complete problem of 3-dimensional matching. Using our main result we establish the NP-completeness of a problem of the best approximation of a symmetric relation on a finite set by an equivalence relation, thus answering in the negative a question proposed implicitly by C.T. Zahn.
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Křivánek, M., Morávek, J. NP-hard problems in hierarchical-tree clustering. Acta Informatica 23, 311–323 (1986). https://doi.org/10.1007/BF00289116
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DOI: https://doi.org/10.1007/BF00289116