Parameterized Approximation Schemes for Clustering with General Norm Objectives

This paper considers the well-studied algorithmic regime of designing a (1+epsilon)-approximation algorithm for a k-clustering problem that runs in time f(k,epsilon)poly(n) (sometimes called an efficient parameterized approximation scheme or EPAS for short). Notable previous results of this kind include EPASes in the high-dimensional Euclidean setting for k-center as well as k-median, and k-means. However, existing EPASes handle only basic objectives (such as k-center, k-median, and k-means) and are tailored to the specific objective and metric space.

Our main contribution is a clean, simple, and unified algorithm that yields an EPAS for a large variety of clustering objectives (for example, k-means, k-center, k-median, priority k-center, l-centrum, ordered k-median, socially fair k-median aka robust k-median, or more generally monotone norm k-clustering) and metric spaces (for example, continuous high-dimensional Euclidean spaces, metrics of bounded doubling dimension, bounded treewidth metrics, and planar metrics), and which is (almost) entirely oblivious to the underlying objective and metric space.

Key to our approach is a new concept that we call bounded epsilon-scatter dimension -- an intrinsic complexity measure of a metric space that is a relaxation of the standard notion of bounded doubling dimension.

This is joint work with Fateme Abbasi, Sandip Banerjee, Jarosław Byrka, Parinya Chalermsook, Ameet Gadekar, Kamyar Khodamoradi, Dániel Marx, and Roohani Sharma.