Abstract
This paper is devoted to the notion of average consensus together with some generalizations involving L p -norms.
We prove that finding one of these consensus dissimilarities out of a profile of dissimilarities is NP-hard for ultrametrics, quasi-utrametrics and proper dissimilarities satisfying the Bertrand and Janowitz k-point inequality. The NP-hardness of finding a consensus dissimilarity for a pyramid (also called an indexed pseudo-hierarchy) is also proved in the case of one of the two possible alternatives for generalized average consensus.
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BANDELT, H.J. and DRESS, A.W.M. (1989): Weak hierarchies associated with dissimilarity measures: an additive clustering technique. Bulletin of Mathematical Biology, 51 (1), 133–166.
BARTHELEMY, J.-P. and JANOWITZ, M.F. (1991): A formal theory of consensus. SIAM J. Discr. Math., 4, 305–322.
BARTHELEMY, J.-P., LECLERC, B. and MONJARDET, B. (1986): On the use of ordered sets in problems of comparison and consensus of classifications. J. of Classification, 3, 187–224.
BARTHELEMY, J.-P. and MONJARDET, B. (1981): The median procedure in cluster analysis and social choice theory. Math. Soc. Sci., 1, 235–267.
BENZECRI, J.-P. (1973): L’analyse des données, Volume 1: Taxonomie. Dunod, Paris.
BERTRAND, P. and JANOWITZ, M.J. (1999): The k-weak hierarchies: an extension of weak hierarchies. preprint.
BOCK, H.H. (1994): Classification and clustering: Problems for the future. In Diday et al. (eds): New approaches in classification and data analysis. Springer-Verlag, Berlin, 3–24.
CHEPOI, V. and FICHET, B. (2000): le-Approximation via subdominants. Journal of Mathematical psychology, to appear.
CUCUMEL, G. (1990): Construction d’une hiérarchie consensus à l’aide d’une ultramétrique centrale. In Recueil des Textes des Présentations du Colloque sur les Méthodes et Domaines d’Applications de la Statistique 1990, Bureau de la Statistique du Québec, Quebec, 235 - 243.
DIATTA, J. and FICHET, B. (1994): From Asprejan hierarchies and Bandelt-Dress weak hierarchies to quasi-hierarchies. In E. Diday et al. (eds.), New approaches in classification and data analysis, Springer Verlag, Berlin, 111–118.
DIATTA, J. and FICHET, B. (1998): Quasi-ultrametriccs and their 2-balls hyper-graphs. Discrete Math., 192, 87–102.
DIDAY, E. (1984): Orders and overlapping clustering by pyramids. Research Report, 73, INRIA.
DIDAY, E. (1986): Orders and overlapping clusters in pyramids. In De Leeuw et al. (ed.): Multidimensional Data Analysis, DSWO Press, Leiden, 201–234.
FICHET, B. (1984): Sur une extension de la notion de hiórarchie et son üquivalence avec une matrice de Robinson. In Journóes de Statistiques de la Grande Motte.
FICHET, B. (1986): Data analysis: Geometric and algebraic structures. In Y.A Pro- horov and V.U. Sasonov (eds.): First world Congress of the Bernoulli Society Proceedings. V.N.U. Science Press, Utrecht, 123–132.
FICHET, B. (1988): 4-spaces in data analysis. In H.H. Bock (ed.): Classification and Related Methods of Data Analysis. Amsterdam: North Holland, 439–444.
GAREY, M.R. and JOHNSON, D.S. (1979): Computer and intractability: A guide in the theory of NP-completeness. Freeman, New York.
GORDON, A. (1980): On the assessment and comparison of classification. In R. Tomassone (ed.): Analyse des données et informatique, INRIA, Le Chesnay, 149–160.
GORDON, A. (1981): Classification methods for the exploratory analysis of multivariate data. Chapman and Hill, London.
GORDON, A. (1986): Consensus supertrees: the synthesis of rooted trees containing overlapping sets of labelled leaves. J. of Classification, 3, 2, 335–348.
JARDINE, J.P.J., JARDINE, N. and SIBSON, R.S. (1967): The structure and construction of taxonomic hierarchies. Mathematical Biosciences, 1, 171–179.
JOHNSON, S.C. (1967): Hierarchical clustering schemes. Psychometrika, 32, 241–254.
KRIVANEK, M. and MORAVEK, J. (1986): NP-hard problems in hierarchical clustering. Acta Informatica, 23, 311–323.
LAPOINTE, F.-J. and CUCUMEL, G. (1997): The average consensus procedure. Combination of weighted trees containing identical or overlapping sets of taxa. Systematic Biology, 46, 306–312.
MARGUSH, T. and MCMORRIS, F.R. (1981): Consensus n-trees. Bull. Math. Biology, 43, 239–224.
MCMORRIS, F.R. (1985): Axioms for consensus functions defined on undirected phylogenetic trees. Math. Biosciences, 74, 77–80.
RÉGNIER, S. (1965): Sur quelques aspects mathématiques des problèmes de classification automatique. ICC Bull., 4, 175–191. Reprinted in Math. Sc. Hum., 82(1983), 13–29.
WAKABAYASHI, Y. (1986): Aggregation of binary relations: algorithmic and polyhedral investigations, Ph.D. Dissertation, University of Augsburg.
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Barthélemy, JP., Brucker, F. (2000). Average Consensus in Numerical Taxonomy and Some Generalizations. In: Gaul, W., Opitz, O., Schader, M. (eds) Data Analysis. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58250-9_8
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DOI: https://doi.org/10.1007/978-3-642-58250-9_8
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