The sum of radii problem ($k$-MSR) asks, given a metric space on $n$ points, to place $k$ balls covering all points so as to minimize the sum of their radii. Despite extensive study from the perspectives of approximation and parameterized algorithms, the exact parameterized complexity of the problem and the existence of efficient parameterized approximation schemes remained open. We advance this understanding on both the hardness and algorithmic fronts. We begin by showing that $k$-MSR is $W[2]$-hard parameterized by $k$, thereby pinpointing its location in the $W$-hierarchy. Moreover, via our reduction, we rule out efficient parameterized approximation schemes (EPAS)--that is, $(1+ε)$-approximations running in time $f(k,ε)\cdot \mathrm{poly}(n)$--unless $W[2] = FPT$. Assuming the Exponential Time Hypothesis, we further rule out such algorithms running in time $f(k,ε)\cdot n^{o(k)}$, strengthening recent lower bounds for the problem. On the algorithmic side, we study $k$-MSR under the framework of mergeable constraints, which captures a broad class of clustering constraints, including fairness, diversity, and lower bounds. We obtain an FPT $(\frac{8}{3}+ε)$-approximation, improving upon the previous best guarantee of $(4+ε)$. Moreover, given access to a suitable assignment subroutine, we achieve a $(2+ε)$-approximation, matching the best known bound for the unconstrained problem. This, in turn, yields $(2+ε)$ FPT-approximations for several important settings, including $(t,k)$-fair, $(α,β)$-fair, $\ell$-diversity, and private clustering.
🌊
