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arxiv: 2605.13759 · v1 · pith:FM357X3Ynew · submitted 2026-05-13 · 💻 cs.LG

Fast and effective algorithms for fair clustering at scale

Claudio Mantuano , Manuel Kammermann , Philipp Baumann This is my paper

Pith reviewed 2026-05-14 19:35 UTC · model grok-4.3

classification 💻 cs.LG
keywords fair clusteringk-meansheuristicsscalabilityfairness constraintsprotected groupsclustering costtrade-off control
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The pith

Heuristics achieve scalable fair k-means clustering by enforcing group representation targets while minimizing sum of squared distances.

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

The paper develops a framework for partitioning data into a fixed number of clusters such that each protected group meets a user-specified minimum representation in every cluster, all while keeping the total within-cluster sum of squared Euclidean distances low. Three heuristics are built on this framework: one that emphasizes solution quality and extra constraints, one that trades some quality for better speed, and one optimized purely for handling millions of objects quickly. A reader would care because clustering now appears in fairness-sensitive settings like customer segmentation or student grouping, where violating group balance can produce systematically biased outcomes and prior methods either ignored scale or lost too much clustering quality.

Core claim

We propose a general framework for fair clustering that provides precise control over the cost-fairness trade-off and introduce three heuristics based on it. The first heuristic focuses on solution quality and the flexibility to incorporate additional constraints, the second improves scalability while retaining high solution quality, and the third is designed for maximum scalability, producing solutions for instances with millions of objects in seconds. The proposed heuristics outperform existing approaches in comprehensive numerical experiments on benchmark datasets.

What carries the argument

A general fair-clustering framework whose core mechanism adds explicit representation constraints for protected groups and solves the resulting trade-off via local-search or relaxation heuristics that adjust cluster assignments to meet fairness targets while reducing sum-of-squared-distance cost.

If this is right

  • Users can set an exact target fairness level and obtain a clustering whose cost is competitive with unconstrained k-means.
  • Datasets containing millions of points become practical for fair clustering because runtimes drop to seconds.
  • The same framework supports additional side constraints without losing the ability to control the cost-fairness balance.
  • Quality and scalability can be traded off by selecting among the three heuristics depending on data size.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may transfer to other partitioning tasks such as facility location or community detection when group-balance requirements are added.
  • In streaming or online settings the third heuristic could serve as a fast warm-start for incremental re-clustering under fairness rules.
  • Empirical success on Euclidean data raises the question whether similar constraint-handling ideas work for non-Euclidean distances or kernelized clustering.

Load-bearing premise

That fairness constraints requiring minimum group representation in each cluster can be satisfied without destroying the geometric structure that makes the sum of squared distances a meaningful clustering objective.

What would settle it

On a moderate-sized benchmark instance whose globally optimal fair clustering solution is known by exhaustive search, the heuristics would return assignments whose total cost exceeds the optimum by more than a few percent or that violate the stated fairness targets.

Figures

Figures reproduced from arXiv: 2605.13759 by Claudio Mantuano, Manuel Kammermann, Philipp Baumann.

Figure 1
Figure 1. Figure 1: (a) synthetic dataset; (b) vanilla clustering optimal solution; (c) fair clustering optimal solution. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Decomposition scheme used by the proposed heuristics. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Applying MPFC to the illustrative example: (a) initialization of cluster centers; (b) assignment of [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Flow network used in each stage l > 1 of the assignment step of MS-FlowFC. stages l ′ = 1, . . . , l − 1. For each previously assigned protected group l ′ , a lower bound can be computed by multiplying the target balance Bs(X, λ) by the number of objects from this protected group assigned to center j, namely |Cj ∩ Gl ′s|. The largest of these lower bounds is the one that must be imposed for protected group… view at source ↗
Figure 5
Figure 5. Figure 5: Applying MS-FlowFC to the illustrative example: (a) initialization of cluster centers and assignment of [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Applying S-MPFC to the illustrative example: (a) generation of [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Boxplots of clustering costs for solutions obtained with different random seeds on the [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Cost-fairness trade-off of O&C (a), MPFC (b), and MS-FlowFC (c) on the [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
read the original abstract

Clustering is an unsupervised machine learning task that consists of identifying groups of similar objects. It has numerous applications and is increasingly used in fairness-sensitive domains where objects represent individuals, such as customers, employees, or students. We address a fair clustering problem in which objects belong to protected groups. The problem consists of partitioning the objects into a predefined number of clusters while attaining a user-defined target level of fairness, meaning that each protected group is sufficiently represented in each cluster. The objective is to minimize the clustering cost, defined as the sum of squared Euclidean distances between the objects and the centers of their clusters. Since clustering cost and fairness are generally in conflict, managing the trade-off between them is essential in practical applications. Existing methods provide limited control over this trade-off and either fail to scale to large datasets or, when they scale, produce low-quality solutions. We propose a general framework for fair clustering that provides precise control over the cost-fairness trade-off and introduce three heuristics based on it. The first heuristic focuses on solution quality and the flexibility to incorporate additional constraints, the second improves scalability while retaining high solution quality, and the third is designed for maximum scalability, producing solutions for instances with millions of objects in seconds. The proposed heuristics outperform existing approaches in comprehensive numerical experiments on benchmark datasets. The source code of our heuristics and instructions for reproducing the experiments are publicly available on GitHub.

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.

Referee Report

2 major / 1 minor

Summary. The paper introduces a general framework for fair k-means clustering that enforces a user-specified minimum representation of protected groups in each cluster while minimizing the sum of squared Euclidean distances to cluster centers. It proposes three heuristics (quality-focused local search, a scalable variant, and a maximum-scalability version) that are claimed to provide precise control over the cost-fairness trade-off, outperform prior methods on benchmark datasets, and scale to instances with millions of points in seconds, with publicly available code.

Significance. If the reported empirical dominance and scalability hold under rigorous validation, the framework and open-source implementation would offer a practical advance for deploying clustering in fairness-sensitive applications such as customer segmentation or educational grouping, where explicit trade-off control is needed.

major comments (2)
  1. [Experiments] The central claim of outperformance rests on the numerical experiments, yet the manuscript provides no statistical significance tests, no comparison against exact solvers on small instances, and no Pareto-front coverage metrics; this is load-bearing because the heuristics lack approximation guarantees and their local-search/repair steps have no proven bound on deviation from globally optimal trade-offs.
  2. [§4] The exact definition of the fairness metric, data-exclusion rules, and interaction between the fairness relaxation/repair steps and the squared-Euclidean objective are not fully specified, preventing verification that reported wins are not artifacts of favorable instance selection or local-optima bias.
minor comments (1)
  1. [Framework] Notation for the fairness target parameter and the precise form of the relaxed objective could be clarified with an explicit equation in the framework section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the major comments point by point below and will revise the manuscript to incorporate the suggested improvements for stronger empirical validation and clarity.

read point-by-point responses

  1. Referee: [Experiments] The central claim of outperformance rests on the numerical experiments, yet the manuscript provides no statistical significance tests, no comparison against exact solvers on small instances, and no Pareto-front coverage metrics; this is load-bearing because the heuristics lack approximation guarantees and their local-search/repair steps have no proven bound on deviation from globally optimal trade-offs.

    Authors: We agree that statistical significance tests, comparisons to exact solvers on small instances, and Pareto-front coverage metrics would strengthen the presentation. In the revised manuscript we will add Wilcoxon signed-rank tests (or paired t-tests where appropriate) across repeated runs and datasets to assess significance of the reported improvements. We will also include a new subsection with experiments on small instances (n ≤ 500) solved to optimality via an ILP formulation using a commercial solver, reporting optimality gaps for our heuristics. Finally, we will report Pareto-front coverage metrics such as the fraction of the cost-fairness curve dominated by each method. While the heuristics are not accompanied by approximation guarantees, the new experiments will provide direct evidence of their practical performance relative to optima on small cases and will be discussed explicitly in the text. revision: yes

  2. Referee: [§4] The exact definition of the fairness metric, data-exclusion rules, and interaction between the fairness relaxation/repair steps and the squared-Euclidean objective are not fully specified, preventing verification that reported wins are not artifacts of favorable instance selection or local-optima bias.

    Authors: We apologize for the insufficient detail in §4. In the revision we will provide a complete mathematical definition of the fairness metric (including the precise minimum-representation thresholds per group and cluster), explicitly list all data-exclusion or preprocessing rules applied to the benchmark datasets, and add a formal description (with pseudocode) of how the relaxation and repair steps interact with the squared-Euclidean objective. These additions will enable independent verification and will clarify that the reported results are not driven by instance selection or local-optima artifacts. revision: yes

Circularity Check

0 steps flagged

No circularity: framework and heuristics are defined independently of results

full rationale

The paper introduces a fair clustering framework parameterized by an explicit user-supplied fairness target and the standard k-means objective (sum of squared Euclidean distances). The three heuristics are constructed via local-search, relaxation, and repair steps whose definitions do not presuppose the claimed performance outcomes. No equation or claim reduces a derived quantity to a fitted parameter or self-citation by construction. Self-citations, if present, are not load-bearing for the central algorithmic definitions or the empirical comparison. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on the standard Euclidean k-means objective and a user-specified fairness target; no new free parameters are introduced beyond the usual cluster count k and the fairness level. No invented entities or non-standard axioms are required.

axioms (2)
  • domain assumption Euclidean distance is an appropriate measure of dissimilarity for the objects being clustered.
    Invoked when defining the clustering cost as sum of squared Euclidean distances.
  • domain assumption The fairness requirement can be expressed as a set of linear or convex constraints on cluster membership proportions.
    Required for the framework to admit efficient heuristics.

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Reference graph

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