Individually Fair Clustering

• Individually Fair Clustering is the study of clustering under fairness constraints that ensure similar individuals are treated similarly across various formulations.
• Methodologies include fair-radius, feature-based co-clustering, service parity, average-dissimilarity, and distributional fairness, each addressing different aspects of equitable treatment.
• Key insights reveal that while these fairness models often lead to NP-hard problems, they also inspire bicriteria approximations and scalable algorithms balancing clustering cost with fairness guarantees.

Individually fair clustering studies clustering under constraints intended to formalize the principle that similar individuals should be treated similarly. In the clustering literature, this principle has not converged to a single canonical definition. Instead, several distinct formulations have been developed: nearest-neighbor coverage constraints based on a point-specific fair radius; feature-based co-clustering constraints defined in a fairness feature space; service-parity constraints in which similar points should receive comparable center distances; average-dissimilarity conditions requiring each point to be closer, on average, to its own cluster than to any other; and distributional formulations in which similar individuals receive statistically similar distributions over centers (Mahabadi et al., 2020, Kar et al., 2021, Chakrabarti et al., 2021, Kleindessner et al., 2020, Anderson et al., 2020). The area therefore combines approximation algorithms, LP rounding, local search, dynamic programming, and specialized extensions for outliers, graph data, and representation learning (Chakrabarty et al., 2021, Bateni et al., 2024, Maity et al., 2024, Maity et al., 7 Oct 2025, Ghodsi et al., 2024, Ruoss et al., 2020).

1. Core formulations

A central formulation defines, for each point $x$ in a set $P$ of size $n$, a fair radius

$r(x) = \min \{r : |B(x, r)| \geq n/k\},$

or equivalently $\delta(x)$ as the radius of the smallest ball around $x$ containing at least $n/k$ points. A clustering with center set $S$ is individually fair when every point has a center within that radius, or within a relaxed radius $\alpha r(x)$: $d(x,S) \leq \alpha \cdot r(x).$ This fair-radius model underlies a large part of the algorithmic literature on individually fair $P$0-median, $P$1-means, and $P$2-center (Mahabadi et al., 2020, Vakilian et al., 2021, Chakrabarty et al., 2021, Bateni et al., 2024).

A different line of work defines fairness through a separate feature space. In "Feature-based Individual Fairness in $P$3-Clustering" (Kar et al., 2021), each point $P$4 has a feature vector $P$5, similarity is defined by

$P$6

two points are $P$7-similar if $P$8, and $P$9 denotes the points $n$0-similar to $n$1. A clustering with assignment $n$2 is $n$3-individually fair if

$n$4

Here fairness is not proximity to a center, but co-membership with sufficiently many similar points.

In "A New Notion of Individually Fair Clustering: $n$5-Equitable $n$6-Center" (Chakrabarti et al., 2021), fairness is defined by parity of service quality relative to a similarity set $n$7. The per-point constraint is

$n$8

and the aggregate variant is

$n$9

This formulation compares the service a point receives to the service received by points it considers similar.

A further formulation, introduced in "A Notion of Individual Fairness for Clustering" (Kleindessner et al., 2020), requires that each point be closer on average to points in its own cluster than to points in any other cluster. If $r(x) = \min \{r : |B(x, r)| \geq n/k\},$0 is the cluster containing $r(x) = \min \{r : |B(x, r)| \geq n/k\},$1, then $r(x) = \min \{r : |B(x, r)| \geq n/k\},$2 is treated individually fair if either $r(x) = \min \{r : |B(x, r)| \geq n/k\},$3, or

$r(x) = \min \{r : |B(x, r)| \geq n/k\},$4

A probabilistic generalization appears in "Distributional Individual Fairness in Clustering" (Anderson et al., 2020). Each individual $r(x) = \min \{r : |B(x, r)| \geq n/k\},$5 is assigned a distribution $r(x) = \min \{r : |B(x, r)| \geq n/k\},$6 over at most $r(x) = \min \{r : |B(x, r)| \geq n/k\},$7 centers, and fairness is imposed through $r(x) = \min \{r : |B(x, r)| \geq n/k\},$8-divergence: $r(x) = \min \{r : |B(x, r)| \geq n/k\},$9 The clustering objective is then the expected $\delta(x)$0-norm cost,

$\delta(x)$1

Formulation	Core fairness requirement	Representative source
Fair-radius coverage	$\delta(x)$2	(Mahabadi et al., 2020)
Feature-based co-clustering	$\delta(x)$3	(Kar et al., 2021)
$\delta(x)$4-equitable service parity	Service within factor $\delta(x)$5 of similar points	(Chakrabarti et al., 2021)
Average-dissimilarity fairness	Own-cluster average distance no worse than any other cluster	(Kleindessner et al., 2020)
Distributional fairness	$\delta(x)$6	(Anderson et al., 2020)

2. Feasibility, hardness, and price of fairness

Several formulations of individually fair clustering are computationally hard even at the level of feasibility. For the average-dissimilarity notion, deciding whether a data set admits an individually fair $\delta(x)$7-clustering is NP-hard, even when $\delta(x)$8 and the dissimilarity is a metric (Kleindessner et al., 2020). For the feature-based model, deciding whether there exists a clustering with cost less than a trivially fair clustering is NP-hard even when only two facilities are allowed; distinguishing between instances with zero and non-zero optimal cost is NP-complete even for $\delta(x)$9; there is no constant-factor approximation for general instances; and even under triangle inequality there is no fully polynomial time approximation scheme unless $x$0 (Kar et al., 2021).

The $x$1-equitable $x$2-center model yields a structural feasibility threshold. For both EqCenter-PP and EqCenter-AG, there exist instances without any feasible solution when $x$3, whereas every instance admits a feasible solution for $x$4. In that sense, $x$5 is the minimally possible threshold for universal feasibility in that model (Chakrabarti et al., 2021).

The literature also studies the cost incurred by imposing fairness. In the $x$6-equitable setting, the Price of Fairness can be arbitrarily large in the worst case, although bounded-Price-of-Fairness guarantees are obtained in some structured cases, including settings with up to $x$7 centers and, for AG, equal similarity radii (Chakrabarti et al., 2021). In the fair-radius literature, standard $x$8-median or $x$9-means solutions can be arbitrarily unfair according to the individual fairness criterion, which motivates fairness-specific algorithm design rather than post hoc auditing of unconstrained clusterings (Mahabadi et al., 2020).

These hardness and feasibility results explain why the field is dominated by bicriteria guarantees, relaxations of the fairness radius, randomized procedures, and restricted settings such as line metrics or fixed centers.

3. Algorithmic paradigms for fair-radius and service-based models

The fair-radius model has been studied through local search, LP rounding, and reductions to constrained facility-location variants. "Individual Fairness for $n/k$0-Clustering" (Mahabadi et al., 2020) augments classical local search for $n/k$1-median and $n/k$2-means by requiring every candidate solution to cover a family of critical balls. The algorithm computes disjoint critical balls $n/k$3, initializes centers so that every critical ball contains a center, and performs local-search swaps of up to 4 centers while preserving this coverage invariant. The resulting guarantees are an $n/k$4-bicriteria approximation for $n/k$5-median and $n/k$6-means, an $n/k$7-approximation for $n/k$8-center, and an $n/k$9-approximation for general $S$0 cost; the number of iterations is $S$1, and the overall runtime is polynomial, specifically $S$2 when swaps of size at most 4 are allowed (Mahabadi et al., 2020).

"Improved Approximation Algorithms for Individually Fair Clustering" (Vakilian et al., 2021) replaces the earlier $S$3-style approximation factors by stronger bicriteria bounds. For any $S$4, it gives a $S$5-bicriteria approximation for $S$6-clustering, a $S$7-bicriteria approximation for $S$8-median, and a $S$9-bicriteria approximation for $\alpha r(x)$0-center. The algorithm constructs disjoint critical regions, reduces the problem to facility location with $\alpha r(x)$1-cost under a matroid constraint, and uses an extended LP-based framework inspired by Charikar et al. and Swamy. The same work also describes the reduction from individually fair clustering to a clustering problem with a group fairness requirement proposed by Kleindessner et al., which is essentially the median matroid problem (Vakilian et al., 2021).

A second LP-based direction appears in "Better Algorithms for Individually Fair $\alpha r(x)$2-Clustering" (Chakrabarty et al., 2021). Its LP has variables $\alpha r(x)$3 for opening centers and $\alpha r(x)$4 for assignments, with the fairness condition enforced explicitly through

$\alpha r(x)$5

The Fair-Round procedure first solves the LP, then filters points using radii

$\alpha r(x)$6

where $\alpha r(x)$7, and finally rounds the resulting representative structure. The paper states an $\alpha r(x)$8-approximation algorithm, with fairness factor $\alpha r(x)$9 and cost factor $d(x,S) \leq \alpha \cdot r(x).$0; the special cases are $d(x,S) \leq \alpha \cdot r(x).$1 for $d(x,S) \leq \alpha \cdot r(x).$2-median and $d(x,S) \leq \alpha \cdot r(x).$3 for $d(x,S) \leq \alpha \cdot r(x).$4-means. It also introduces a sparsification technique that reduces LP size while preserving bounded degradation in fairness and cost (Chakrabarty et al., 2021).

"A Scalable Algorithm for Individually Fair K-means Clustering" (Bateni et al., 2024) emphasizes practical scale. It gives the first fast local-search algorithm running in $d(x,S) \leq \alpha \cdot r(x).$5 time with an $d(x,S) \leq \alpha \cdot r(x).$6 bicriteria approximation for individually fair $d(x,S) \leq \alpha \cdot r(x).$7-means. The method uses anchor points and anchor zones, modified greedy seeding, and constrained swaps that preserve coverage of each anchor zone. The design goal is explicitly to retain theoretical guarantees while scaling beyond the sizes handled by earlier local-search or LP-based baselines (Bateni et al., 2024).

For the $d(x,S) \leq \alpha \cdot r(x).$8-equitable $d(x,S) \leq \alpha \cdot r(x).$9-center objective, combinatorial algorithms are also available. Given optimal value $P$00 and $P$01, the main guarantee is

$P$02

which becomes a 10-approximation under the structured condition $P$03. The same work notes that assigning each point to its nearest center does not always respect the fairness constraints, and therefore develops a non-trivial iterative assignment procedure (Chakrabarti et al., 2021).

4. Feature-based and distributional individual fairness

The feature-based formulation separates the space used to define fairness from the space used to optimize clustering cost. In "Feature-based Individual Fairness in $P$04-Clustering" (Kar et al., 2021), the algorithm LP-FAIR first selects initial centers using a standard $P$05-clustering routine that ignores fairness, then solves a linear relaxation over fixed centers: $P$06 subject to

$P$07

The solution is rounded by assigning each point $P$08 to facility $P$09 with probability $P$10, repeating the procedure $P$11 times and returning the lowest-cost solution. The main theorem states that for any $P$12, if $P$13 for all $P$14, then with high probability the algorithm returns cost at most $P$15, where $P$16 is the LP optimum, and each point $P$17 has at least $P$18 similar points in its assigned cluster (Kar et al., 2021).

This model is notable because the fairness features may be arbitrary and may be disjoint from clustering features. The paper therefore departs from the nearest-neighbor fair-radius tradition: fairness is expressed by peer co-location rather than by center accessibility. The same framework can be composed with a black-box vanilla clustering algorithm to obtain an overall fair clustering algorithm when distances satisfy triangle inequality (Kar et al., 2021).

The distributional model in "Distributional Individual Fairness in Clustering" (Anderson et al., 2020) replaces deterministic assignment by a distribution $P$19 over centers for each point $P$20. The fair assignment problem is expressed as a convex program: $P$21 subject to

$P$22

Its algorithm ALG-IF first runs a $P$23-approximation for standard clustering to obtain centers $P$24, then solves the fair-assignment LP for these fixed centers. The theorem gives

$P$25

The same framework extends to combined group fairness and individual fairness within protected groups by adding constraints

$P$26

together with within-group individual fairness constraints. The paper also observes that enforcing individual fairness globally can force all points to be assigned uniform distributions, and for that reason advocates within-group enforcement in the combined setting (Anderson et al., 2020).

5. Special settings and extensions

The average-dissimilarity formulation admits an exact algorithm on the real line. For one-dimensional Euclidean data, "A Notion of Individual Fairness for Clustering" (Kleindessner et al., 2020) proves that for any $P$27 there exists a fair $P$28-clustering with contiguous clusters, and gives a dynamic programming algorithm with runtime $P$29. Its recurrence is presented as

$P$30

with fairness checks reduced to cluster boundaries in the contiguous setting (Kleindessner et al., 2020).

Outlier-aware individually fair clustering has become a separate subarea. "Linear Programming based Approximation to Individually Fair $P$31-Clustering with Outliers" (Maity et al., 2024) extends fair-radius $P$32-means to allow a set $P$33 of outliers with $P$34, excluded from both cost and fairness. The LP introduces variables $P$35, $P$36, and $P$37: $P$38 subject to

$P$39

After solving the LP, the OutRound procedure thresholds outliers and the FairRound algorithm of Negahbani and Chakrabarty rounds the remaining fractional solution. The paper states

$P$40

followed by final guarantees $P$41 for $P$42-means, $P$43 for $P$44-median, and a 16-approximation to the fair radius for all inliers (Maity et al., 2024).

"Local Search-based Individually Fair Clustering with Outliers" (Maity et al., 7 Oct 2025) studies the same setting using a randomized local-search framework. It adopts the Jung et al. fairness requirement $P$45 for non-outlier points, allows a relaxation $P$46, and introduces BaseCent to identify fairness-based outliers before running LSFO with constrained local search. The method discards outliers in two stages—fairness-based and cost-based—and proves an $P$47-approximation to the cost of the optimal $P$48-fair clustering. The total number of outliers is bounded by

$P$49

and the runtime is

$P$50

A key lemma states that if the $P$51 points are covered by $P$52 $P$53-anchor zones, then a set of $P$54 $P$55-anchor zones can cover $P$56 points (Maity et al., 7 Oct 2025).

Individual fairness has also been extended beyond point-set clustering. "Towards Cohesion-Fairness Harmony: Contrastive Regularization in Individual Fair Graph Clustering" (Ghodsi et al., 2024) proposes iFairNMTF, which minimizes

$P$57

where $P$58 is the Laplacian of a contrastive matrix built from same-group and different-group indicators. The second term is a contrastive fairness regularizer, and $P$59 controls the cohesion-fairness trade-off (Ghodsi et al., 2024).

Representation learning offers another route. "Learning Certified Individually Fair Representations" (Ruoss et al., 2020) learns an encoder $P$60 such that

$P$61

For clustering, the paper states that if a clustering algorithm does not split $P$62-balls of radius $P$63, then similar individuals will always be assigned to the same cluster. This is a sufficient condition rather than a clustering-specific optimization criterion, but it places individually fair clustering in a broader certified-fairness pipeline (Ruoss et al., 2020).

6. Empirical findings and conceptual relations

Empirical studies consistently report that fairness-aware methods can improve fairness substantially while remaining competitive on clustering objectives, although the precise trade-off depends on the fairness notion. In the fair-radius local-search study, the proposed algorithm improves clustering cost over FairKCenter by factors $P$64–$P$65 for $P$66-median and $P$67–$P$68 for $P$69-means, while increasing the fairness ratio by factors $P$70–$P$71 and $P$72–$P$73, respectively (Mahabadi et al., 2020). In the feature-based setting, LP-FAIR is reported to be 12.5% fairer on average than the fairest baseline and 34.5% lower in normalized cost on average than the best cost baseline, with more than 95.5% of points satisfying the feature-based fairness constraint across datasets (Kar et al., 2021).

Practical scalability has become an explicit benchmark. The scalable local-search algorithm LSPP runs on datasets up to 581,012 points in less than 2 hours and empirically attains fairness bound ratios in the range $P$74–$P$75, whereas vanilla $P$76-means can violate the fair-radius bound by factors of 60–90 (Bateni et al., 2024). The LP-based Fair-Round algorithm reports that clustering cost is often within 1% of the LP lower bound and never more than 15%, that the maximum fairness violation is at most 1.27 in the reported experiments, and that more than 80% of points are perfectly fair; sparsification lowers runtime by orders of magnitude (Chakrabarty et al., 2021).

Outlier-aware studies report similar patterns. The LP-based IFXO method yields lower cost than clustering without outlier removal and outperforms isolation forest plus fair clustering baselines in both cost and fairness metrics (Maity et al., 2024). The local-search-based LSFO method typically attains the lowest $P$77-means cost among compared methods, with fairness violation ratio $P$78 often between $P$79 and $P$80, practical outlier counts typically less than $P$81 when $P$82 of $P$83, and stable behavior under parameter changes (Maity et al., 7 Oct 2025).

The literature also documents several important non-equivalences. Group-fair clustering is not necessarily individually fair: this is shown both in the average-dissimilarity line of work and in the distributional framework, where standard group-fair algorithms may satisfy demographic constraints while violating a large fraction of individual-fairness constraints (Kleindessner et al., 2020, Anderson et al., 2020). Conversely, individual fairness can imply group fairness only under additional conditions; in the distributional model, the bound

$P$84

links group-level deviation to an earthmover-distance condition on the protected group’s spread relative to the whole population (Anderson et al., 2020).

Taken together, these results show that individually fair clustering is not a single optimization problem but a family of related problems that differ in what counts as similar individuals, what counts as similar treatment, and whether fairness is enforced through deterministic assignments, center accessibility, service parity, co-clustering, or probabilistic output distributions. The main technical trajectory of the field has therefore been to turn each fairness notion into a tractable surrogate: critical-ball coverage, matroid-constrained facility location, LP-based fair assignment, dynamic programming on structured domains, and outlier-aware local search.