Clustering to Minimize Cluster-Aware Norm Objectives

We initiate the study of the following general clustering problem. We seek to partition a given set $P$ of data points into $k$ clusters by finding a set $X$ of $k$ centers and assigning each data point to one of the centers. The cost of a cluster, represented by a center $x\in X$, is a monotone, symmetric norm $f$ (inner norm) of the vector of distances of points assigned to $x$. The goal is to minimize a norm $g$ (outer norm) of the vector of cluster costs. This problem, which we call $(f,g)$-Clustering, generalizes many fundamental clustering problems such as $k$-Center, $k$-Median , Min-Sum of Radii, and Min-Load $k$-Clustering . A recent line of research (Chakrabarty, Swamy [STOC'19]) studies norm objectives that are oblivious to the cluster structure such as $k$-Median and $k$-Center. In contrast, our problem models cluster-aware objectives including Min-Sum of Radii and Min-Load $k$-Clustering. Our main results are as follows. First, we design a constant-factor approximation algorithm for $(\textsf{top}_\ell,\mathcal{L}_1)$-Clustering where the inner norm ($\textsf{top}_\ell$) sums over the $\ell$ largest distances. Second, we design a constant-factor approximation\ for $(\mathcal{L}_\infty,\textsf{Ord})$-Clustering where the outer norm is a convex combination of $\textsf{top}_\ell$ norms (ordered weighted norm).