An LP-based k-means algorithm for balancing weighted point sets

For example, while finite termination of all k-means variants for unweighted point sets is a simple consequence of the existence of only finitely many partitions of a given set of points, the situation is more involved for weighted point sets, as there are infinitely many partial membership clusterings. Using polyhedral theory, we show that the number of iterations of weight-balanced k-means is bounded above by nO(dk), so in particular it is polynomial for fixed k and d. This is similar to the known worst-case upper bound for classical k-means for unweighted point sets and unrestricted cluster sizes, despite the much more general framework. We conclude with the discussion of some additional favorable properties of our method.