FPT Constant-Approximations for Capacitated Clustering to Minimize the Sum of Cluster Radii

Clustering with capacity constraints is a fundamental problem that attracted significant attention throughout the years. In this paper, we give the first FPT constant-factor approximation algorithm for the problem of clustering points in a general metric into k clusters to minimize the sum of cluster radii, subject to non-uniform hard capacity constraints. In particular, we give a (15+ε)-approximation algorithm that runs in 20(k2log k)· n3 time. When capacities are uniform, we obtain the following improved approximation bounds: A (4 + ε)-approximation with running time 2O(klog(k/ε))n3, which significantly improves over the FPT 28-approximation of Inamdar and Varadarajan [ESA 2020]; a (2 + ε)-approximation with running time 2O(k/ε2 ·log(k/ε))dn3 and a (1+ε)-approximation with running time 2O(kdlog ((k/ε)))n3 in the Euclidean space; and a (1 + ε)-approximation in the Euclidean space with running time 2O(k/ε2 ·log(k/ε))dn3 if we are allowed to violate the capacities by (1 + ε)-factor. We complement this result by showing that there is no (1 + ε)-approximation algorithm running in time f(k)· nO(1), if any capacity violation is not allowed.