An approximation algorithm for the uniform capacitated k-means problem

In this paper, we consider the uniform capacitated k-means problem (UC-k-means), an extension of the classical k-means problem (k-means) in machine learning. In the UC-k-means, we are given a set $$\mathcal {D}$$ D of n points in d-dimensional space and an integer k. Every point in the d-dimensional space has an uniform capacity which is an upper bound on the number of points in $$\mathcal {D}$$ D that can be connected to this point. Every two-point pair in the space has an associated connecting cost, which is equal to the square of the distance between these two points. We want to find at most k points in the space as centers and connect every point in $$\mathcal {D}$$ D to some center without violating the capacity constraint, such that the total connecting costs is minimized. Based on the technique of local search, we present a bi-criteria approximation algorithm, which has a constant approximation guarantee and violates the cardinality constraint within a constant factor, for the UC-k-means.