# Clustering with or without the approximation

## Abstract

We study algorithms for clustering data that were recently proposed by Balcan et al. (SODA’09: 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1068–1077, 2009a) and that have already given rise to several follow-up papers. The input for the clustering problem consists of points in a metric space and a number *k*, specifying the desired number of clusters. The algorithms find a clustering that is provably close to a target clustering, provided that the instance has the “(1+*α*,*ε*)-property”, which means that the instance is such that all solutions to the *k*-median problem for which the objective value is at most (1+*α*) times the optimal objective value correspond to clusterings that misclassify at most an *ε* fraction of the points with respect to the target clustering. We investigate the theoretical and practical implications of their results.

Our main contributions are as follows. First, we show that instances that have the (1+*α*,*ε*)-property and for which, additionally, the clusters in the target clustering are large, are easier than general instances: the algorithm proposed in Balcan et al. (SODA’09: 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1068–1077, 2009a) is a constant factor approximation algorithm with an approximation guarantee that is better than the known hardness of approximation for general instances. Further, we show that it is *NP*-hard to check if an instance satisfies the (1+*α*,*ε*)-property for a given (*α*,*ε*); the algorithms in Balcan et al. (SODA’09: 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1068–1077, 2009a) need such *α* and *ε* as input parameters, however. We propose ways to use their algorithms even if we do not know values of *α* and *ε* for which the assumption holds. Finally, we implement these methods and other popular methods, and test them on real world data sets. We find that on these data sets there are no *α* and *ε* so that the dataset has both (1+*α*,*ε*)-property and sufficiently large clusters in the target solution. For the general case where there are no assumptions about the cluster sizes, we show that on our data sets the performance guarantee proved by Balcan et a. (SODA’09: 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1068–1077, 2009a) is meaningless for the values of *α*,*ε* for which the data set has the (1+*α*,*ε*)-property. The algorithm nonetheless gives reasonable results, although it is outperformed by other methods.

## Keywords

Clustering*k*-median Algorithms Approximation

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