arxiv: 2605.06398 · v1 · submitted 2026-05-07 · 💻 cs.DS

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

Ameet Gadekar

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classification 💻 cs.DS

keywords sum of radii clusteringparameterized complexityW[2]-hardnessapproximation algorithmsmergeable constraintsfair clusteringdiversity constraints

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The pith

The k-sum-of-radii clustering problem is W[2]-hard parameterized by k, ruling out efficient parameterized approximation schemes, while mergeable constraints permit an FPT (8/3 + ε)-approximation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper resolves the parameterized complexity of the minimum sum-of-radii clustering problem with k clusters by proving it W[2]-hard. The hardness implies there is no (1+ε)-approximation algorithm running in time f(k, ε) times a polynomial in n, unless W[2] equals FPT. Under the exponential time hypothesis, even algorithms running in n to the power o(k) are ruled out. For clustering problems with mergeable constraints, which include fairness and diversity requirements, the authors develop an FPT algorithm achieving an approximation ratio of 8/3 + ε, better than the previous 4 + ε guarantee, and a 2 + ε ratio when an assignment subroutine is available.

Core claim

We prove that k-MSR is W[2]-hard parameterized by k and that this rules out efficient parameterized approximation schemes unless W[2] = FPT. Assuming ETH we rule out n^{o(k)} time EPAS. For mergeable constraints we give an FPT (8/3 + ε)-approximation algorithm improving on (4 + ε), and a (2 + ε) FPT-approximation given an assignment subroutine, which yields results for (t,k)-fair, (α,β)-fair, ℓ-diversity, and private clustering.

What carries the argument

The mergeable constraints framework that captures a broad class of clustering constraints including fairness, diversity, and lower bounds, allowing the combination of feasible solutions while maintaining feasibility for improved approximation guarantees.

Load-bearing premise

The W[2]-hardness reduction and the approximation algorithms both assume that the mergeable-constraint framework correctly models real-world clustering constraints without introducing gaps in the reduction or analysis.

What would settle it

The discovery of a (1 + ε)-approximation algorithm for k-MSR that runs in time f(k, ε) · n^{O(1)} would falsify the claim that no such EPAS exists unless W[2] = FPT.

Figures

Figures reproduced from arXiv: 2605.06398 by Ameet Gadekar.

**Figure 2.** Figure 2: Derivation of our algorithmic results. Overview. Our first goal is to compute a low-cost feasible cover for an instance of k-MSR under a mergeable constraint. Towards this, in Section 3.1, we design an FPT algorithm (GBC) that computes a ball cover B that is Π∗ -respecting, with cost at most (2 + ε) times the optimum for some optimal solution Π∗ . Using B, we then present in Section 3.2 an FPT algorithm (F… view at source ↗

**Figure 3.** Figure 3: An example of a 2-cover in which the minimum cost is 4/3 of the sum of radii of the balls. Consider this iteration. By Theorem 7, there exists a cover B ∈ B that is Π∗ -respecting and satisfies cost(B) ≤ (2 + ε)OPT(I). Applying Theorem 8, there exists F ∈ F such that F is a Π∗ -feasible cover and cost(F) ≤ (2 + ε)OPT(I). Since F is a feasible cover, we can apply Lemma 2 to obtain a feasible clustering (C, … view at source ↗

read the original 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, $(1+\epsilon)$-approximations running in time $f(k,\epsilon)\cdot \mathrm{poly}(n)$--unless $W[2] = FPT$. Assuming the Exponential Time Hypothesis, we further rule out such algorithms running in time $f(k,\epsilon)\cdot n^{o(k)}$, 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 $(\frac{8}{3}+\epsilon)$-approximation, improving upon the previous best guarantee of $(4+\epsilon)$. Moreover, given access to a suitable assignment subroutine, we achieve a $(2+\epsilon)$-approximation, matching the best known bound for the unconstrained problem. This, in turn, yields $(2+\epsilon)$ FPT-approximations for several important settings, including $(t,k)$-fair, $(\alpha,\beta)$-fair, $\ell$-diversity, and private clustering.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper settles the W[2]-hardness of k-MSR and rules out EPAS, while lifting the approximation ratio for mergeable constraints from 4+ε to 8/3+ε.

read the letter

The core news is that k-MSR is W[2]-hard parameterized by k, and the same reduction shows there is no EPAS unless W[2] = FPT. Under ETH the lower bound strengthens to rule out f(k,ε) n^{o(k)} time. On the algorithmic side the authors give an FPT (8/3 + ε)-approximation for the mergeable-constraint version of the problem and recover the (2 + ε) bound once an assignment oracle is supplied. Both results are new relative to the cited prior work. The hardness part cleanly locates the problem in the W-hierarchy and blocks the obvious route to an FPT PTAS. The approximation part extends the unconstrained (2 + ε) guarantee to a useful family of fairness and diversity constraints without blowing up the running time. The mergeable framework itself is presented as a natural abstraction that covers several existing constrained-clustering models, which is a practical plus. The reduction and the algorithmic techniques appear to rest on standard parameterized tools rather than ad-hoc fitting, so the claims do not look circular. One minor soft spot is that the precise definition of “mergeable” will need careful checking in the full text to confirm it really captures the intended applications without hidden restrictions. The oracle result is conditional, but the paper is explicit about that. This is a focused contribution for people working on parameterized approximation algorithms for geometric clustering. Anyone tracking the exact complexity of covering problems or looking for FPT approximations under fairness constraints will get direct value. The open question it resolves and the concrete ratio improvement are enough to justify sending the paper to a serious referee rather than a desk reject.

Referee Report

0 major / 2 minor

Summary. The manuscript shows that k-MSR is W[2]-hard parameterized by k, ruling out EPAS unless W[2]=FPT, and under ETH no f(k,ε) n^{o(k)} time algorithms. For mergeable constraints, an FPT (8/3+ε)-approximation is given, improving on (4+ε), and with assignment oracle a (2+ε)-approximation is achieved, which applies to fair, diverse, and private clustering settings.

Significance. This work provides a clear placement of k-MSR in the W-hierarchy and rules out certain approximation schemes, which is important for the field. The mergeable constraints framework is a positive contribution as it unifies several constraints. The approximation results are improvements and match known bounds in some cases, making them significant for both theory and practice in constrained clustering.

minor comments (2)

The previous best guarantee of (4+ε) should be referenced with a citation in the abstract or introduction for completeness.
Clarify how the mergeable property is verified for the listed applications like (t,k)-fair clustering in the applications section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of our work and for recommending minor revision. We are glad that the placement of k-MSR in the W-hierarchy, the ruling out of EPAS, and the improvements to (8/3+ε) and (2+ε) approximations under mergeable constraints (including applications to fair, diverse, and private clustering) are viewed as significant contributions.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claims rest on a standard W[2]-hardness reduction from a known parameterized-hard problem and on new FPT approximation algorithms constructed via dynamic programming and other standard techniques within the mergeable-constraints framework. No derivation step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the hardness and approximation results are independently derived and externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard parameterized-complexity assumptions and the definition of mergeable constraints; no free parameters or new entities are introduced.

axioms (2)

domain assumption W[2] ≠ FPT and the Exponential Time Hypothesis hold.
Invoked to obtain the hardness and time lower bounds for EPAS.
domain assumption Mergeable constraints admit a suitable assignment subroutine that can be called in FPT time.
Required for the (2+ε) approximation result.

pith-pipeline@v0.9.0 · 5627 in / 1497 out tokens · 33168 ms · 2026-05-08T05:44:30.920819+00:00 · methodology

discussion (0)

Reference graph

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