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 ∊ X, is a monotone, symmetric norm f (called inner norm) of the vector of distances of points assigned to x. The goal is to minimize a norm g (called 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 (i.e., (𝓛_∞, 𝓛_∞)-Clustering), k-Median (i.e., (𝓛₁, 𝓛₁)-Clustering), Min-Sum of Radii (i.e., (𝓛_∞, 𝓛₁)-Clustering), and Min-Load k-Clustering (i.e., (𝓛₁, L_∞)-Clustering). A recent line of research (Byrka et al. [STOC’18], Chakrabarty, Swamy [ICALP’18, STOC’19], and Abbasi et al. [FOCS’23]) 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 (top_ℓ, 𝓛₁)- Clustering where the inner norm (top_ℓ) sums over the ℓ largest distances. This unifies (up to constant factors) the best known results for k-Median and Min-Sum of Radii. Second, we design a constant-factor approximation for (𝓛_∞, Ord)-Clustering where the outer norm is a convex combination of top_ℓ norms (ordered weighted norm). This generalizes known results for k-Center and Min-Sum of Radii. Obtaining a constant-factor approximation for more general settings that include (𝓛₁, 𝓛_∞)-Clustering (Min-Load k-Clustering) seems challenging because even an o (k )-approximation is unknown for this problem. We can still use our two main results to obtain first (although non-constant) approximations for these problems including general monotone, symmetric norms.

Our algorithm for (top_ℓ, 𝓛₁)-Clustering relies on a reduction to a novel generalization of k-Median, which we call Ball k-Median. In this problem, we aim at selecting k balls (rather than k centers) and pay for connecting the points to these balls as well as for the (scaled) radii of the balls. To obtain a constant-factor approximation for this problem we unify various algorithmic techniques originally designed for the cluster-oblivious k-Median objective (Jain and Vazirani [JACM 2001], Li and Svensson [STOC’13]) and for the cluster-aware Min-Sum of Radii Objective (Charikar and Panigrahi [STOC’01] and Ahmadian and Swamy [ICALP’16]).

^* Martin Herold is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) — Project number 399223600. This work is part of the project TIPEA that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 850979). We are grateful to an anonymous reviewer for making concrete suggestions how to substantially simplify our algorithm for (Top, 𝓛₁ (-Clustering by applying the techniques of Ahmadian, Swamy [4].