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The pheromone trail is reinforced for those objects that belong to the same class. We present some preliminary results, compared to results of other techniques, such as simulated annealing, genetic algorithms, tabu search, and k-means. Our results are as good as the best of the above methods. 1 Introduction The partitioning problem is well known for its problems with local minima. Indeed, most of the methods for clustering with a fixed number of classes find local optima of the criterion to be optimized, such as k-means or its variants (dynamical clusters, transfers, Isodata (Diday et al.

14 Fionn Murtagh 17. Ogielski, A. T. and Stein, D. L. (1985). "Dynamics of Ultrametric Spaces," Physical Review Letters, 55, 1634-1637. 18. Parisi, G. and Ricci-Tersenghi, F. (2000). "On the Origin of Ultrametricity," Journal of Physics A: Mathematical and General, 33, 113-129. 19. , Angles d'Auriac, J. , and Doucot, B. (1985). "On the Degree of Ultrametricity," Le Journal de Physique-Lettres, 46, L-945-L-952. 20. A. (1986). "Ultrametricity for Physicists," Reviews of Modern Physics, 58, 765-788.

PM associated to each ant Run k-means on each pm in order to converge to a local minimum of W for t = 1 to t max do: for m = 1 to M, do S times: Choose at random an object i Choose an object j according to (7) Assign j to the class of i end-for Calculate B(pl), ... , B(pM) and keep the best value Update I according to (3), (4) and (5) end-for. Notice that the algorithm has the following parameters: the number M of ants, the initial value of pheromone 10, the maximum number of iterations tmax' weight a of the pheromone in (7), weight f3 of the local heuristic in (7), the evaporation coefficient p in (3), and the number S of transfers for each ant.