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

Abstract: 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 $n<sup>{O(k)}$. We also provide $(1+\epsilon)$ approximation algorithms for both MSR and MSD with running-times of $O(kn) +(1/\epsilon)<sup>{O(dk)}$ 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 $\alpha&gt;1$. For $\alpha$-MSD we prove an exponential time ETH-based lower bound for $\alpha&gt;\log 3$. All algorithms can also be modified to handle outliers. Moreover, we can extend the results to variants that observe fairness constraints, as well as to the general framework of 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 $n<sup>{O(k)}$ time is tight for MSR and $\alpha$-MSR even in doubling spaces, and that $2<sup>{o(k)}$ bounds are impossible for MSD.