Dimension-Free Parameterized Approximation Schemes for Hybrid Clustering

Summary

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{dist(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+ε, 1+ε)-bicrtieria approximation that runs in time 2^{(kd/ε)^{O(1)}} ⋅ n^{O(1)} for inputs in ℝ^d; such a bicriteria solution uses balls of radius (1+ε)r instead of r, and has a cost at most 1+ε 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 ε - 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.