The Bane of Low-Dimensionality Clustering

In this paper, we give a conditional lower bound of n on running time for the classic k-median and k-means clustering objectives (where n is the size of the input), even in lowdimensional Euclidean space of dimension four, assuming the Exponential Time Hypothesis (ETH). We also consider k-median (and k-means) with penalties where each point need not be assigned to a center, in which case it must pay a penalty, and extend our lower bound to at least three-dimensional Euclidean space. This stands in stark contrast to many other geometric problems such as the traveling salesman problem, or computing an independent set of unit spheres. While these problems benefit from the so-called (limited) blessing of dimensionality, as they can be solved in time n ) or 2 1−1/d in d dimensions, our work shows that widely-used clustering objectives have a lower bound of n, even in dimension four. We complete the picture by considering the two-dimensional case: we show that there is no algorithm that solves the penalized version in time less than n √ , and provide a matching upper bound of n √ . The main tool we use to establish these lower bounds is the placement of points on the moment curve, which takes its inspiration from constructions of point sets yielding Delaunay complexes of high complexity. ∗The project leading to this application has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 748094. The work of A. de Mesmay is partially supported by the French ANR project ANR-16-CE40-0009-01 (GATO). The work of A. Roytman is partially supported by Thorup’s Advanced Grant DFF-0602-02499B from the Danish Council for Independent Research.