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.

翻译：半径和问题（$k$-MSR）要求：给定$n$个点上的度量空间，放置$k$个覆盖所有点的球，使得这些球半径之和最小。尽管从近似算法和参数化算法的角度已有广泛研究，但该问题的精确参数化复杂性以及高效参数化近似方案的存在性仍未解决。我们在困难性和算法性两方面推进了这一理解。首先，我们证明$k$-MSR关于参数$k$是$W[2]$-困难的，从而确定了其在$W$-层次结构中的位置。此外，通过我们的归约，我们排除了存在高效参数化近似方案（EPAS）的可能性——即运行时间为$f(k,ε)\cdot \mathrm{poly}(n)$的$(1+ε)$-近似算法——除非$W[2] = FPT$。在指数时间假设下，我们进一步排除了运行时间为$f(k,ε)\cdot n^{o(k)}$的此类算法，强化了针对该问题的近期下界。在算法方面，我们在可合并约束框架下研究$k$-MSR，该框架涵盖了广泛的聚类约束，包括公平性、多样性和下界约束。我们获得了FPT的$(\frac{8}{3}+ε)$-近似算法，改进了此前$(4+ε)$的最佳保证。此外，在能够调用合适的分配子例程的情况下，我们实现了$(2+ε)$-近似算法，达到了无约束问题已知的最佳界。这进而为若干重要场景（包括$(t,k)$-公平、$(α,β)$-公平、$\ell$-多样性和私有聚类）提供了$(2+ε)$的FPT近似算法。