Constant-Factor Approximations for Doubly Constrained Fair k-Center, k-Median and k-Means

Annika Hennes; Johanna Hillebrand; Nicole Funk; Sarah Sturm

arxiv: 2604.16061 · v1 · submitted 2026-04-17 · 💻 cs.DS · cs.CY· cs.LG

Constant-Factor Approximations for Doubly Constrained Fair k-Center, k-Median and k-Means

Nicole Funk , Annika Hennes , Johanna Hillebrand , Sarah Sturm This is my paper

Pith reviewed 2026-05-10 07:34 UTC · model grok-4.3

classification 💻 cs.DS cs.CYcs.LG

keywords fair clusteringk-centerk-mediank-meansapproximation algorithmsgroup fairnessdiverse centersmetric spaces

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The pith

Algorithms achieve constant-factor approximations for k-clustering under both group fairness and diverse center selection.

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

The paper develops approximation algorithms for discrete k-clustering problems in metric spaces that must obey two fairness conditions simultaneously. Group fairness requires each cluster to contain balanced proportions of protected attributes within explicit lower and upper bounds. Diverse center selection requires the chosen centers themselves to come from each attribute group in specified numbers. Prior work combined separate approximations and obtained only an 8-approximation for k-center. The new work improves the k-center guarantee to 4 with a small additive violation and supplies the first constant-factor algorithms for k-median and k-means.

Core claim

We study discrete k-clustering in metric spaces under two fairness constraints: group fairness requiring balanced attribute proportions in clusters and diverse center selection requiring balanced attributes among the chosen centers. We give a 4-approximation for k-center with small additive violation on group fairness and constant-factor approximations for k-median and k-means. The algorithms transform a diverse-center solution into a doubly constrained one using an LP-based approach, and the results generalize to other center-selection constraints such as matroids and knapsacks.

What carries the argument

LP-based transformation that takes any solution satisfying only the diverse-center constraint and produces a clustering that also meets the group-fairness bounds while multiplying the objective by only a constant factor.

Load-bearing premise

The linear-programming adjustment step preserves a constant approximation factor when it enforces the group-fairness bounds on top of an already diverse set of centers.

What would settle it

A concrete metric instance in which every clustering that satisfies both constraints has objective value more than any fixed constant times the objective of the best solution that satisfies only diverse center selection.

read the original abstract

We study discrete k-clustering problems in general metric spaces that are constrained by a combination of two different fairness conditions within the demographic fairness model. Given a metric space (P,d), where every point in P is equipped with a protected attribute, and a number k, the goal is to partition P into k clusters with a designated center each, such that a center-based objective function is minimized and the attributes are fairly distributed with respect to the following two fairness concepts: 1) group fairness: We aim for clusters with balanced numbers of attributes by specifying lower and upper bounds for the desired attribute proportions. 2) diverse center selection: Clusters have natural representatives, i.e., their centers. We ask for a balanced set of representatives by specifying the desired number of centers to choose from each attribute. Dickerson, Esmaeili, Morgenstern and Zhang (2023) denote the combination of these two constraints as doubly constrained fair clustering. They present algorithms whose guarantees depend on the best known approximation factors for either of these problems. Currently, this implies an 8-approximation with a small additive violation on the group fairness constraint. For k-center, we improve this approximation factor to 4 with a small additive violation. This guarantee also depends on the currently best algorithm for DS-fair k-center given by Jones, Nguyen and Nguyen (2020). For k-median and k-means, we propose the first constant-factor approximation algorithms. Our algorithms transform a solution that satisfies diverse center selection into a doubly constrained fair clustering using an LP-based approach. Furthermore, our results are generalizable to other center-selection constraints, such as matroid k-clustering and knapsack constraints.

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

They give the first constant-factor approximations for doubly constrained fair k-median and k-means, plus a 4-approximation for k-center.

read the letter

The main thing to know is that this paper delivers the first constant-factor approximations for k-median and k-means under the combination of group fairness bounds and diverse center selection, while improving the k-center guarantee from 8 to 4 with only a small additive violation on the group constraint. They start from an existing diverse-center solution and use an LP to enforce the proportion bounds on protected attributes, then round or reassign to produce the final clustering. The same LP step is shown to work for matroid and knapsack center constraints as well. This is a direct and useful extension of the prior separate results on each fairness notion. The approach avoids simply multiplying the individual approximation factors, which is the cleanest part of the contribution. The LP transformation is the technical core, and the paper claims it preserves a constant overall. The potential soft spot is whether the objective increase from the LP adjustment and any follow-up rounding stays bounded by a small constant in general metrics, especially for the sum objectives. In k-median or k-means, satisfying the proportion constraints can force some points into distant clusters, and the charging argument or dual analysis needs to control that tightly; the abstract does not spell out the explicit constant or dependence on the additive violation, so the full proofs will need checking. This work is aimed at people already working on approximation algorithms for fair clustering. Readers who know the Jones et al. diverse-center results and the Dickerson et al. doubly constrained setup will get the most from it. The algorithmic ideas are concrete and build on existing machinery, so the paper deserves peer review.

Referee Report

2 major / 1 minor

Summary. The paper studies doubly constrained fair k-clustering (group fairness via lower/upper attribute proportion bounds plus diverse center selection) in general metrics. It improves the prior 8-approximation for k-center to a 4-approximation with small additive group-fairness violation (building on Jones et al. 2020 for the DS-fair subproblem) and gives the first constant-factor approximations for k-median and k-means. The algorithms obtain a DS-fair solution and then apply an LP-based transformation to enforce the group-fairness bounds while preserving a constant-factor guarantee on the objective; the approach is claimed to generalize to matroid and knapsack center-selection constraints.

Significance. If the LP transformation and its rounding analysis indeed produce only a constant-factor blowup in objective value, the results would be significant: they supply the first constant-factor algorithms for the sum objectives under both constraints simultaneously and offer a modular reduction that could extend to other center-selection side constraints. The improvement from 8 to 4 for k-center is also a concrete advance.

major comments (2)

[Abstract] Abstract and the description of the LP-based transformation: the central claim that transforming a DS-fair solution via LP yields a doubly constrained solution whose objective remains within a constant factor of the DS-fair cost (and hence of OPT) is load-bearing for all stated approximation factors. The provided abstract does not exhibit the explicit charging argument, dual bound, or cost-scaling analysis that would guarantee the increase is bounded by a constant independent of the instance, especially for k-median/k-means where a single distant reassignment can add linear cost. Without this, the constant-factor guarantees cannot be verified.
[Abstract] The dependence on prior DS-fair approximations (Jones et al. for k-center; unspecified for median/means) must be shown to compose cleanly with the LP step; if the LP solution is obtained from an approximate DS-fair input, the overall ratio is the product of the two factors, and any additive violation must be tracked through the transformation.

minor comments (1)

[Abstract] The abstract states the results are 'generalizable' to matroid and knapsack constraints but does not indicate whether the same LP formulation and constant factors carry over directly or require additional arguments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. The concerns focus on clarity in the abstract regarding the LP transformation analysis and the composition of approximation factors. We will revise the manuscript to address these points directly.

read point-by-point responses

Referee: [Abstract] Abstract and the description of the LP-based transformation: the central claim that transforming a DS-fair solution via LP yields a doubly constrained solution whose objective remains within a constant factor of the DS-fair cost (and hence of OPT) is load-bearing for all stated approximation factors. The provided abstract does not exhibit the explicit charging argument, dual bound, or cost-scaling analysis that would guarantee the increase is bounded by a constant independent of the instance, especially for k-median/k-means where a single distant reassignment can add linear cost. Without this, the constant-factor guarantees cannot be verified.

Authors: The abstract provides a high-level summary of the results, while the explicit charging argument, dual LP bound, and cost-scaling analysis appear in the technical sections of the full manuscript. We agree that the abstract should more clearly indicate how the LP rounding controls cost increases (including for sum objectives) to preserve a constant factor independent of the instance. We will revise the abstract to include a concise reference to this analysis and the metric properties that bound reassignments. revision: yes
Referee: [Abstract] The dependence on prior DS-fair approximations (Jones et al. for k-center; unspecified for median/means) must be shown to compose cleanly with the LP step; if the LP solution is obtained from an approximate DS-fair input, the overall ratio is the product of the two factors, and any additive violation must be tracked through the transformation.

Authors: We will revise the abstract and introduction to explicitly name the DS-fair approximation factors used for each objective (Jones et al. for k-center, and the relevant constant-factor DS-fair results for k-median/k-means) and to state that the overall guarantee is the product of the DS-fair factor and the constant from the LP transformation. The manuscript already shows that the LP step accepts any DS-fair input and produces a doubly constrained solution while preserving the additive group-fairness violation bound; we will add a short paragraph clarifying this composition and tracking of the additive term. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithms compose external DS-fair approximations with a new LP transformation

full rationale

The derivation begins with a solution satisfying only diverse center selection, obtained via the cited external algorithm of Jones et al. (2020). An LP-based transformation is then applied to enforce the additional group-fairness proportion bounds. No equation, parameter fit, or self-citation in the abstract or described chain reduces the claimed 4-approximation (k-center) or constant-factor guarantees (k-median/k-means) to a quantity defined by the paper's own inputs or prior self-work. The LP step is presented as a novel, independent technique whose analysis (if correct) supplies the constant-factor bound without presupposing the target result. This is the normal case of a paper whose central contribution rests on external benchmarks and standard rounding rather than self-referential definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters or invented entities are introduced. The work relies on the standard assumption that the input is a metric space.

axioms (1)

domain assumption The input forms a metric space satisfying the triangle inequality.
Required for all center-based clustering objectives and for the cited approximation algorithms to apply.

pith-pipeline@v0.9.0 · 5625 in / 1325 out tokens · 70381 ms · 2026-05-10T07:34:00.934904+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Fair range k-center.CoRR, abs/2207.11337, 2022

URL:http://proceedings.mlr.press/v119/mahabadi20a.html. 27 Huy Le Nguyen, Thy Dinh Nguyen, and Matthew Jones. Fair range k-center.CoRR, abs/2207.11337, 2022.arXiv:2207.11337,doi:10.48550/arXiv.2207.11337. 28 James B. Orlin. Max flows in o(nm) time, or better. In Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors,Symposium on Theory of Computing Conf...

work page doi:10.48550/arxiv.2207.11337 2022
[2]

∑ j∈Px′ ij≥1for alli∈CDS 3.y ′is integral. Proof

work page
[3]

For (LP-1.1), considerj∈P. ∑ i∈CDS x′ ij = ∑ i∈CDS   ∑ p∈N(i) xpi∑ ℓ∈CDS :p∈N(ℓ)xpℓ xpj + ∑ p∈θ−1(i) xpj   = ∑ p∈N(CDS) ∑ i∈CDS :p∈N(i) xpi∑ ℓ∈CDS :p∈N(ℓ)xpℓ xpj + ∑ i∈CDS ∑ p∈θ−1(i) xpj = ∑ p∈N(CDS) xpj + ∑ p∈P\N(CDS) xpj = ∑ p∈P xpj = 1, 22 Constant-Factor Approximations for Doubly Constrained Fairk-Clustering where the last equality holds because(x...

work page

[1] [1]

Fair range k-center.CoRR, abs/2207.11337, 2022

URL:http://proceedings.mlr.press/v119/mahabadi20a.html. 27 Huy Le Nguyen, Thy Dinh Nguyen, and Matthew Jones. Fair range k-center.CoRR, abs/2207.11337, 2022.arXiv:2207.11337,doi:10.48550/arXiv.2207.11337. 28 James B. Orlin. Max flows in o(nm) time, or better. In Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors,Symposium on Theory of Computing Conf...

work page doi:10.48550/arxiv.2207.11337 2022

[2] [2]

∑ j∈Px′ ij≥1for alli∈CDS 3.y ′is integral. Proof

work page

[3] [3]

For (LP-1.1), considerj∈P. ∑ i∈CDS x′ ij = ∑ i∈CDS   ∑ p∈N(i) xpi∑ ℓ∈CDS :p∈N(ℓ)xpℓ xpj + ∑ p∈θ−1(i) xpj   = ∑ p∈N(CDS) ∑ i∈CDS :p∈N(i) xpi∑ ℓ∈CDS :p∈N(ℓ)xpℓ xpj + ∑ i∈CDS ∑ p∈θ−1(i) xpj = ∑ p∈N(CDS) xpj + ∑ p∈P\N(CDS) xpj = ∑ p∈P xpj = 1, 22 Constant-Factor Approximations for Doubly Constrained Fairk-Clustering where the last equality holds because(x...

work page