Title: The Bane of Low-Dimensionality Clustering,
Abstract: In this paper, we give a conditional lower bound of $n^{\Omega(k)}$ on
running time for the classic k-median and k-means clustering objectives (where
n is the size of the input), even in low-dimensional 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 $n^{O(k^{1-1/d})}$ or
$2^{n^{1-1/d}}$ in d dimensions, our work shows that widely-used clustering
objectives have a lower bound of $n^{\Omega(k)}$, 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
$n^{o(\sqrt{k})}$, and provide a matching upper bound of $n^{O(\sqrt{k})}$.
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.