A balanced k-means algorithm for weighted point sets

The classical $k$-means algorithm for partitioning $n$ points in $\mathbb{R}^d$ into $k$ clusters is one of the most popular and widely spread clustering methods. The need to respect prescribed lower bounds on the cluster sizes has been observed in many scientific and business applications. In this paper, we present and analyze a generalization of $k$-means that is capable of handling weighted point sets and prescribed lower and upper bounds on the cluster sizes. We call it weight-balanced $k$-means. The key difference to existing models lies in the ability to handle the combination of weighted point sets with prescribed bounds on the cluster sizes. This imposes the need to perform partial membership clustering, and leads to significant differences. 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 $n^{O(dk)}$, 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.