On the Parameterized Approximability of (Mergeable) Sum of Radii Clustering

2026-05-07

AI Summary

This study investigates the computational complexity and algorithmic design for the k-minimum sum-of-radii (k-MSR) clustering problem in metric spaces under the framework of parameterized approximation. It establishes, for the first time, that k-MSR is W[2]-hard with respect to the parameter k, thereby ruling out the existence of an efficient parameterized approximation scheme unless FPT = W[2]. Furthermore, a conditional lower bound is provided under the Exponential Time Hypothesis. For k-MSR under composable constraints—such as fairness, diversity, or privacy—the paper presents two fixed-parameter tractable (FPT) approximation algorithms achieving approximation ratios of 8/3 + ε and 2 + ε, respectively. The latter matches the best-known guarantee for the unconstrained version, thus extending state-of-the-art results to broad constrained settings.

Abstract

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.

Problem

• Sum of Radii Clustering
• Parameterized Complexity
• Approximation Algorithms
• Mergeable Constraints
• k-MSR

Innovation

• parameterized approximation
• sum of radii clustering
• W[2]-hardness
• mergeable constraints
• FPT algorithm