Dimension-Free Parameterized Approximation Schemes for Hybrid Clustering

Hybrid $k$-Clustering is a model of clustering that generalizes two of the most widely studied clustering objectives: $k$-Center and $k$-Median. In this model, given a set of $n$ points $P$, the goal is to find $k$ centers such that the sum of the $r$-distances of each point to its nearest center is minimized. The $r$-distance between two points $p$ and $q$ is defined as $\max\{d(p, q)-r, 0\}$ -- this represents the distance of $p$ to the boundary of the $r$-radius ball around $q$ if $p$ is outside the ball, and $0$ otherwise. This problem was recently introduced by Fomin et al. [APPROX 2024], who designed a $(1+\varepsilon, 1+\varepsilon)$-bicrtieria approximation that runs in time $2^{(kd/\varepsilon)^{O(1)}} \cdot n^{O(1)}$ for inputs in $\mathbb{R}^d$; such a bicriteria solution uses balls of radius $(1+\varepsilon)r$ instead of $r$, and has a cost at most $1+\varepsilon$ times the cost of an optimal solution using balls of radius $r$. In this paper we significantly improve upon this result by designing an approximation algorithm with the same bicriteria guarantee, but with running time that is FPT only in $k$ and $\varepsilon$ -- crucially, removing the exponential dependence on the dimension $d$. This resolves an open question posed in their paper. Our results extend further in several directions. First, our approximation scheme works in a broader class of metric spaces, including doubling spaces, minor-free, and bounded treewidth metrics. Secondly, our techniques yield a similar bicriteria FPT-approximation schemes for other variants of Hybrid $k$-Clustering, e.g., when the objective features the sum of $z$-th power of the $r$-distances. Finally, we also design a coreset for Hybrid $k$-Clustering in doubling spaces, answering another open question from the work of Fomin et al.