FPT Constant Approximation Algorithms for Colorful Sum of Radii

Gregory Gutin; Shuilian Liu; Yicheng Xu; Yong Zhang

REVIEW 3 major objections 3 minor 23 references

FPT Constant Approximation Algorithms for Colorful Sum of Radii

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper establishes the first constant-factor approximation algorithms for the colorful sum of radii problem that run in fixed-parameter tractable time: a (2+ε)-approximation and a (7+ε)-approximation.

desk verdict Both main theorems have load-bearing proof gaps, so the claimed FPT constant-factor approximations for colorful sum of radii are not established; the problem and high-level approach are promising but need major repair. read the letter →

arxiv 2506.13191 v1 pith:OIUYXTUK submitted 2025-06-16 cs.CG cs.DS

Shuilian Liu , Gregory Gutin , Yicheng Xu , Yong Zhang This is my paper · ORCID

classification cs.CGcs.DS MSC 68W2568Q2568U05

keywords ColorfulSumofRadiiOutlier-RobustClusteringConstantApproximationFixed-ParameterTractablek-centerFairAlgorithms

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper targets the colorful sum of radii problem: partition a point set into color classes, choose k centers, and assign each non-outlier point to a center while allowing at most m_i outliers from class i, minimizing the sum of cluster radii. The authors claim the first constant-factor approximation algorithms for this problem that run in fixed-parameter tractable (FPT) time, with the exponential cost placed on the parameters k and m rather than on n. Concretely, they give a (2+ε)-approximation whose running time is exponential in both k and m, and a (7+ε)-approximation whose running time is exponential only in k, obtained by plugging a 3-approximation for colorful k-center into a general framework. The (2+ε) algorithm also applies directly to the classical outlier-robust sum of radii problem. This matters because it converts a problem with only a logarithmic approximation guarantee into one solvable to within a small constant, while keeping polynomial dependence on the data size.

What carries the argument

The load-bearing objects are the near-optimal radius profile and the iterative covering loop. Lemma 1 computes a small candidate set of radius vectors (r̃1,...,r̃k) with r_i* ≤ r̃_i ≤ (1+ε)r_i* by geometric discretization of the interval between a β-approximate k-center value and the optimal cost. In the (2+ε) algorithm, the profile is padded with m zero radii so outliers are treated as zero-radius balls; random point selection plus a guessed radius produces a ball of radius 2r̃_j that contains the entire optimal cluster, and the probability bound (k+m)^{-(k+m)} per profile drives the repetition count. In the (7+ε) algorithm, Lemmas 5-6 and Corollary 7 link colorful k-center to colorful sum of radii by bounding the optimal residual k-center radius by 2r_i*, so a β-approximate subroutine returns radius at most 2βr̃_i; the event E that the assignment σ_i routes the i-th optimal center to the right ball occurs with probability at least (k+m)^{-k}, and the enlarged radius (2β+1)r̃_i covers the cluster while the cost telescopes to (2β+1)(1+ε)OPT.

What would settle it

In a metric with k well-separated singleton clusters and zero default outlier allowance, set n much larger than k and run Algorithm 1's random sampling; the actual probability that k uniform samples hit all k clusters is k!(k/n)^k, far below 1/k^k, so an instance large enough makes the (k+m)^{k+m} repetition bound insufficient for constant success probability. For Algorithm 3, take k-1 well-separated clusters plus a final cluster whose center lies inside an already chosen ball, with outlier budgets tight; compute the optimal radius of the residual (k-i)-center instance and check whether it exceeds 2r_i* — the lemma's claim fails when it does.

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Extended reading notes

Core claim

The central claim is that the colorful sum of radii admits constant-factor approximations in FPT time. The first algorithm guesses a near-optimal radius profile by discretizing the interval from the optimal k-center radius to the optimal sum-of-radii cost, pads the profile with m zero radii for outliers, and repeatedly samples a point, draws an unused radius, and covers the point's optimal cluster with a ball of twice that radius; repeating the random process (k+m)^{k+m} times per profile yields constant success probability and total cost at most (2+ε)OPT. The second algorithm is a generic reduction: whenever a β-approximation for colorful k-center is available, iteratively peel off the largest remaining optimal cluster by running the k-center routine on the residual instance and covering the cluster with a ball of radius (2β+1) times the guessed radius, giving a (2β+1+ε)-approximation; with β=3 the result is 7+ε and the time bound is O($nω^{2}$((k+m)log(k/ε))^k). The paper presents these as the first constant-factor FPT approximations for the problem.

Load-bearing premise

The argument depends on the assumption that randomly sampling points will hit every optimal cluster within the allowed (k+m) draws with probability at least (k+m)^{-(k+m)}, and that after peeling off the largest clusters, the remaining points can be k-centered with radius at most twice the next optimal sum-of-radii radius.

Editorial extensions

If this is right

For any instance with k centers and m total outliers, a (2+ε)-approximation is computable in time O(n^2 log^k(k/ε)(k+m)^{k+m}).
Any future β-approximation for colorful k-center automatically yields a (2β+1+ε)-approximation for colorful sum of radii via Theorem 3.
With the current best 3-approximation for colorful k-center, the colorful sum of radii is (7+ε)-approximable in O(nω^2((k+m)log(k/ε))^k) time.
The outlier-robust sum of radii problem (the ω=1 case) inherits the (2+ε) FPT approximation.
The problem moves from a logarithmic approximation ratio to a small constant while remaining fixed-parameter tractable in k and m.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the sampling analysis were tightened, the (2+ε) algorithm could likely avoid the (k+m)^{k+m} repetition factor, since the real bottleneck is hitting small clusters; a deterministic greedy cover might replace the random draws entirely.
Improving the colorful k-center approximation below 3 would immediately lower the 7+ε ratio, since the framework's guarantee is 2β+1.
The same covering-plus-residual framework should transfer to other sum-type objectives such as sum of diameters, wherever a k-center subroutine with per-color outlier control exists.
The two algorithms indicate a tradeoff curve: exponential dependence on m buys a better ratio (2+ε), while k-only dependence costs a larger constant (7+ε); closing this gap is a natural next step.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Both main theorems fail as written. The problem is real—colorful sum of radii with per-class outlier bounds is a natural variant—and the idea of bootstrapping a sum-of-radii approximation from a colorful k-center subroutine is worth exploring. But the proofs have load-bearing errors that a careful reader will hit immediately.

Theorem 2's analysis assumes that picking a random point and then a random radius from a profile has success probability at least 1/(k+m) per iteration. That would be true if the point were uniform over k clusters and m outliers, but it is uniform over all n points. A singleton cluster is sampled with probability 1/n, not 1/(k+m), so the claimed (k+m)^{k+m} repetitions do not yield constant success probability. The algorithm might be repairable with a different sampling scheme, but that is not what is proved.

Theorem 4 rests on Lemma 6, which has a center-count mismatch. At iteration i, after covering i−1 of the k optimal clusters, the residual instance still contains points from k−i+1 clusters, yet the lemma bounds the optimal largest radius for an instance with only k−i centers. For well-separated clusters with tight outlier budgets, k−i centers cannot cover all remaining clusters, so the bound is false. Corollary 7 and Lemma 8 inherit the problem, and the (2β+1+ε) guarantee collapses. Algorithm 3 also uses σ_i(c*_i), which presumes knowledge of the optimal centers; without an explicit guessing scheme, the pseudocode is not executable.

On the positive side, the paper gives a clear formal setup and a sensible literature review. The running time of the second algorithm, exponential in k but not in m, is a worthwhile target. But the current proofs do not establish any of the advertised results. I would reject. If the authors can fix the sampling analysis and the residual-instance counting, the paper could become a real contribution, but that requires substantial work.

For your reading group: skip it unless you want an exercise in catching subtle parameter errors. I would not send this to a serious referee in its current state; it needs major revision and re-proofing first.

Referee Report

3 major / 3 minor

Summary. The manuscript studies the colorful sum of radii problem, in which points are partitioned into color classes and each class has a per-class outlier bound. The goal is to choose k centers and assign points to clusters while leaving at most m_i points from class P_i unassigned, minimizing the sum of cluster radii. The paper claims the first constant-factor FPT approximation algorithms for this problem: a (2+ε)-approximation running in time O(n^2 log^k(k/ε)(k+m)^{k+m}) via an iterative random-covering scheme, and a (7+ε)-approximation running in time O(nω^2((k+m)log(k/ε))^k) by iteratively invoking a colorful k-center subroutine.

Significance. If correct, the results would be a meaningful contribution: the colorful sum of radii generalizes outlier-robust sum of radii, and constant-factor FPT algorithms with separate parameter dependence on k and on m are natural targets. The paper uses external subroutines (e.g., the Jia et al. colorful k-center algorithm) and does not fit free parameters to data, so the circularity concern is low. However, both main theorems contain load-bearing gaps in their probability and center-count analyses, and the errors are structural rather than cosmetic. The significance of the claimed contribution is therefore not established by the current manuscript.

major comments (3)

[Section 3, Theorem 2 proof and Algorithm 1] The success probability analysis omits the probability of sampling a point from each optimal cluster. In each of the k+m selections in Lines 9-10, the proof conditions on having selected the correct radius but does not account for the fact that the uniformly chosen point must belong to the appropriate optimal cluster. A singleton optimal cluster is hit with probability only 1/|P| in a uniform draw, and the probability of hitting all k optimal clusters and the m outlier positions can be as small as (k+m)!/n^{k+m} or worse. Repeating the procedure (k+m)^{k+m} times per profile therefore does not give a constant success probability, and the claimed (2+ε)-approximation guarantee of Theorem 2 is unsupported.
[Section 4.1, Lemma 6 and Corollary 7] The center-count mismatch is load-bearing. At the start of iteration i, the algorithm has already covered clusters C*_1,...,C*_{i-1}, so the residual point set P' still contains points from k-i+1 optimal clusters C*_i,...,C*_k. Lemma 6 asserts that the residual instance I'=(P',d,k-i,m') has optimal largest radius at most 2r*_i, but in general k-i centers cannot cover k-i+1 well-separated clusters, and the per-class outlier bounds m' do not allow discarding an entire cluster. The proof's statement that all remaining points can be covered by one ball of radius at most 2r*_i is false when the remaining optimal clusters are far apart. Consequently Corollary 7 and Lemma 8 fail, and the (2β+1+ε) cost analysis of Theorem 3, and hence Theorem 4, is not supported.
[Section 4.1, Algorithm 2] The Counting algorithm contains an implementation error. In Line 4, inside the loop over covered points, it sets ρ'_i ← ρ_i + 1, which adds the original requirement ρ_i for every covered point instead of incrementing by one. Then, in Line 8, the loop overwrites the result with max{0, ρ_i}, so the function returns the unupdated requirement rather than the number of points still required to be covered. This makes the computation of the residual outlier vector m' in Algorithm 3 incorrect, independently of the center-count issue in Lemma 6.

minor comments (3)

[Section 3, Remark 1] The derandomization remark is incomplete: it states that the entire random process can be derandomized by enumerating all possible radii, but Line 9 of Algorithm 1 also chooses points uniformly from a shrinking set P, and the remark does not explain how those choices are enumerated or bounded.
[Section 4.1, Lemma 6] The proof of Lemma 6 refers to Figure 4.1, but no figure appears in the manuscript text; the caption alone is insufficient to verify the claimed covering argument.
[Throughout] The notation for complexity bounds is inconsistent, e.g., O(log^k_{1+ε}(k/ε)) appears in Lemma 1 while the abstract and Theorem 2 use O(log^k(k/ε)); the base of the logarithm should be stated clearly everywhere.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: the FPT approximation guarantees are derived from radius-profile guessing and an external colorful k-center subroutine, not from the target result.

full rationale

The paper's central results do not reduce to their inputs by construction. Lemma 1 guesses the optimal radius profile by discretizing intervals derived from a beta-approximate colorful k-center solution; this is an enumeration over candidate profiles, not a fit of the target sum-of-radii cost. Theorem 2's covering argument uses random point choices and a guessed radius profile, while Theorem 3 and Theorem 4 reduce the colorful sum of radii to a beta-approximation algorithm for colorful k-center (Jia et al. [21], an external result). No parameter is fitted to a subset of the data and then renamed as a prediction. The only self-citation, Chen et al. [10], appears in the introduction as background on FPT algorithms for the plain sum of radii and is not used to justify Theorem 2, Theorem 3, or Theorem 4. The flagged correctness concerns in the paper—the unaccounted point-selection probability in Theorem 2 and the center-count mismatch in Lemma 6/Corollary 7—are genuine proof gaps but not circularity: they do not make a claimed prediction equal to an input by definition or reduce the result to a self-citation chain. The circularity burden is therefore low.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no fitted constants and no invented entities. It relies on standard metric axioms, an external colorful k-center subroutine, and a radius-profile discretization lemma whose per-coordinate guarantee is not exact for very small radii.

assumptions (3)

standard math Metric space triangle inequality
Used throughout the covering arguments, e.g., Lemma 6 and Lemma 9.
domain assumption Existence of a β-approximation algorithm for colorful k-center
Theorem 3 assumes a subroutine Col-Cen with ratio β; the paper uses the 3-approximation from Jia et al. [21]. This is an external result, not derived here.
ad hoc to paper Radius profile discretization (Lemma 1)
The lemma assumes the largest optimal radius is bracketed by a β-approximate k-center value and that smaller radii lie in [ε/k r*_1, r*_1]; the per-coordinate guarantee is not exact for very small radii.

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Cite this review

Pith. "Pith review of FPT Constant Approximation Algorithms for Colorful Sum of Radii." pith.science (2026). https://pith.science/paper/OIUYXTUK

@misc{pith2026250613191,
  author       = {Pith},
  title        = {Pith review of: FPT Constant Approximation Algorithms for Colorful Sum of Radii},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OIUYXTUK}},
  note         = {Machine review of arXiv:2506.13191}
}

abstract

We study the colorful sum of radii problem, where the input is a point set $P$ partitioned into classes $P_1, P_2, \dots, P_\omega$, along with per-class outlier bounds $m_1, m_2, \dots, m_\omega$, summing to $m$. The goal is to select a subset $\mathcal{C} \subseteq P$ of $k$ centers and assign points to centers in $\mathcal{C}$, allowing up to $m_i$ unassigned points (outliers) from each class $P_i$, while minimizing the sum of cluster radii. The radius of a cluster is defined as the maximum distance from any point in the cluster to its center. The classical (non-colorful) version of the sum of radii problem is known to be NP-hard, even on weighted planar graphs. The colorful sum of radii is introduced by Chekuri et al. (2022), who provide an $O(\log \omega)$-approximation algorithm. In this paper, we present the first constant-factor approximation algorithms for the colorful sum of radii running in FPT (fixed-parameter tractable) time. Our contributions are twofold: We design an iterative covering algorithm that achieves a $(2+\varepsilon)$-approximation with running time exponential in both $k$ and $m$; We further develop a $(7+\varepsilon)$-approximation algorithm by leveraging a colorful $k$-center subroutine, improving the running time by removing the exponential dependency on $m$.

Figures

Figures reproduced from arXiv: 2506.13191 by the authors.

**Figure 4.1.** Given that the first i−1 optimal clusters are covered, the inclusion of c ∗ i within an existing ball ensures that all remaining points are coverable within radius 2r ∗ i Lemma 5, we conclude that the optimal radius of the colorful (k − i)-center problem on I ′ is also at most 2r ∗ i , i.e., ¯r ∗ i ≤ 2r ∗ i . □ The colorful k-center provides a useful upper bound of 2r ∗ i for covering the i-th largest cluster in the… view at source ↗

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Clustering what matters: Optimal approximation for clustering with outliers

Akanksha Agrawal, Tanmay Inamdar, Saket Saurabh, and Jie Xue. Clustering what matters: Optimal approximation for clustering with outliers. Journal of Artificial Intelligence Research , 78:143–166, 2023. 14

work page 2023

[2] [2]

A technique for obtaining true approximations for k-center with covering constraints

Georg Anegg, Haris Angelidakis, Adam Kurpisz, and Rico Zenklusen. A technique for obtaining true approximations for k-center with covering constraints. Mathematical Programming, 192(1):3–27, 2022

work page 2022

[3] [3]

A constant approximation for colorful k-center

Sayan Bandyapadhyay, Tanmay Inamdar, Shreyas Pai, and Kasturi Varadarajan. A constant approximation for colorful k-center. In Pro- ceedings of the 27th Annual European Symposium on Algorithms (ESA), volume 144, pages 1–14, 2019

work page 2019

[4] [4]

Fair algorithms for clustering

Suman Bera, Deeparnab Chakrabarty, Nicolas Flores, and Maryam Ne- gahbani. Fair algorithms for clustering. Advances in Neural Information Processing Systems, 32:4955–4966, 2019

work page 2019

[5] [5]

Moritz Buchem, Katja Ettmayr, Hugo K. K. Rosado, and Andreas Wiese. A (3 + ε)-approximation algorithm for the minimum sum of radii problem with outliers and extensions for generalized lower bounds. In Proceedings of the 35th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1738–1765. SIAM, 2024

work page 2024

[6] [6]

Fpt approximations for fair k-min-sum-radii

Lena Carta, Lukas Drexler, Annika Hennes, Clemens R¨ osner, and Melanie Schmidt. Fpt approximations for fair k-min-sum-radii. In Pro- ceedings of the 35th International Symposium on Algorithms and Com- putation (ISAAC), volume 322, pages 1–18, 2024

work page 2024

[7] [7]

Algorithms for facility location problems with outliers

Moses Charikar, Samir Khuller, David M Mount, and Giri Narasimhan. Algorithms for facility location problems with outliers. In Proceed- ings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 642–651, 2001

work page 2001

[8] [8]

Clustering to minimize the sum of cluster diameters

Moses Charikar and Rina Panigrahy. Clustering to minimize the sum of cluster diameters. Journal of Computer and System Sciences, 68(2):417– 441, 2004

work page 2004