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 $\text{eps}>0$, for any instance with $n$ input points and for any positive integer $n^{\mathrm{\prime }}\le n$, we compute in polynomial time a clustering of at least $(1-\text{eps})n^{\mathrm{\prime }}$ points of cost at most a constant factor greater than the optimal cost of clustering $n^{\mathrm{\prime }}$ points. The approximation guarantee grows with $\frac{1}{\text{eps}}$. 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).