FPT approximations for Capacitated Sum of Radii and Diameters
<|reference_start|>FPT approximations for Capacitated Sum of Radii and Diameters: The Capacitated Sum of Radii problem involves partitioning a set of points $P$, where each point $p\in P$ has capacity $U_p$, into $k$ clusters that minimize the sum of cluster radii, such that the number of points in the cluster centered at point $p$ is at most $U_p$. We begin by showing that the problem is APX-hard, and that under gap-ETH there is no parameterized approximation scheme (FPT-AS). We then construct a $\approx5.83$-approximation algorithm in FPT time (improving a previous $\approx7.61$ approximation in FPT time). Our results also hold when the objective is a general monotone symmetric norm of radii. We also improve the approximation factors for the uniform capacity case, and for the closely related problem of Capacitated Sum of Diameters.<|reference_end|>
@article{filtser2024fpt,
    title={FPT approximations for Capacitated Sum of Radii and Diameters},
    author={Arnold Filtser and Ameet Gadekar},
    journal={arXiv preprint arXiv:2409.04984},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.04984},
    primaryClass={cs.DS}
}