Abstract
ConsiderN entities to be classified, with given weights, and a matrix of dissimilarities between pairs of them. The split of a cluster is the smallest dissimilarity between an entity in that cluster and an entity outside it. The single-linkage algorithm provides partitions intoM clusters for which the smallest split is maximum. We consider the problems of finding maximum split partitions with exactlyM clusters and with at mostM clusters subject to the additional constraint that the sum of the weights of the entities in each cluster never exceeds a given bound. These two problems are shown to be NP-hard and reducible to a sequence of bin-packing problems. A Θ (N 2) algorithm for the particular caseM =N of the second problem is also presented. Computational experience is reported.
Résumé
SoientN objets à classifier, avec des poids donnés, et une matrice de dissimilarités entre paires de ces objets. L'écart d'une classe est la plus petite dissimilarité entre un objet de cette classe et un objet en dehors d'elle. L'algorithme du lien simple foumit des partitions enM classes dont le plus petit écart est maximum. Nous étudions comment obtenir des partitions d'écart maximum en exactementM classes ou en au plusM classes sous la contrainte additionnelle que la somme des poids des objets de chaque classe ne dépasse jamais une borne donnée. Nous montrons que ces deux problèmes sont NP-difficiles et réductibles à une séquence de problèmes de mise-en-boites (bin-packing). Nous proposons aussi un algorithme en Θ (N 2) pour le cas particulierM =N du second problème. Enfin, nous présentons des résultats de calcul.
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Acknowledgments: Work of the first author was supported in part by AFOSR grants 0271 and 0066 to Rutgers University and was done in part during a visit to GERAD, Ecole Polytechnique de Montréal, whose support is gratefully acknowledged. Work of the second and third authors was supported by NSERC grant GP0036426 and by FCAR grant 89EQ4144. We are grateful to Silvano Martello and Paolo Toth for making available to us their program MTP for the bin-paking problem and to three anonymous referees for comments which helped to improve the presentation of the paper.
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Hansen, P., Jaumard, B. & Musitu, K. Weight constrained maximum split clustering. Journal of Classification 7, 217–240 (1990). https://doi.org/10.1007/BF01908717
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DOI: https://doi.org/10.1007/BF01908717