Improved fixed-parameter bounds for Min-Sum-Radii and Diameters $k$-clustering and their fair variants

We provide improved upper and lower bounds for the Min-Sum-Radii (MSR) and Min-Sum-Diameters (MSD) clustering problems with a bounded number of clusters $k$. In particular, we propose an exact MSD algorithm with running-time $n^{O(k)}$. We also provide $(1+\epsilon)$ approximation algorithms for both MSR and MSD with running-times of $O(kn) +(1/\epsilon)^{O(dk)}$ in metrics spaces of doubling dimension $d$. Our algorithms extend to $k$-center, improving upon previous results, and to $\alpha$-MSR, where radii are raised to the $\alpha$ power for $\alpha>1$. For $\alpha$-MSD we prove an exponential time ETH-based lower bound for $\alpha>\log 3$. All algorithms can also be modified to handle outliers. Moreover, we can extend the results to variants that observe \emph{fairness} constraints, as well as to the general framework of \emph{mergeable} clustering, which includes many other popular clustering variants. We complement these upper bounds with ETH-based lower bounds for these problems, in particular proving that $n^{O(k)}$ time is tight for MSR and $\alpha$-MSR even in doubling spaces, and that $2^{o(k)}$ bounds are impossible for MSD.