Min-sum clustering (with outliers)

We give a constant factor polynomial time pseudo-approximation algorithm for min-sum clustering with or without outliers. The algorithm is allowed to exclude an arbitrarily small constant fraction of the points. For instance, we show how to compute a solution that clusters 98% of the input data points and pays no more than a constant factor times the optimal solution that clusters 99% of the input data points. More generally, we give the following bicriteria approximation: For any ϵ > 0, for any instance with n input points and for any positive integer n^′ ≤ n, we compute in polynomial time a clustering of at least (1 − ϵ)n^′ points of cost at most a constant factor greater than the optimal cost of clustering n^′ points. The approximation guarantee grows with ¹_ϵ. Our results apply to instances of points in real space endowed with squared Euclidean distance, as well as to points in a metric space, where the number of clusters, and also the dimension if relevant, is arbitrary (part of the input, not an absolute constant).