Welfare-Centric Clustering Framework

• Welfare-centric clustering is a paradigm that optimizes group welfare by jointly balancing distance metrics and proportional representation in cluster assignments.
• It integrates two key objectives—Rawlsian to protect the worst-off group and Utilitarian to minimize overall group disutility—offering a nuanced fairness framework.
• The approach employs a two-stage algorithm with approximative center selection and LP-based rounding, ensuring near-optimal and equitable clustering outcomes.

Welfare-centric clustering is a clustering paradigm in which the optimization target is group welfare rather than overall distortion or a fairness constraint appended to a conventional clustering objective. In the formulation developed in "Welfare-Centric Clustering" (Zhang et al., 14 Aug 2025), welfare is modeled through group-level disutility that jointly captures the distances incurred by group members and the extent to which cluster composition departs from target proportional representation. The framework therefore differs both from representation-based fair clustering, which emphasizes proportional mixing, and from socially fair clustering, which equalizes or protects against high group-specific clustering cost. It is naturally expressed through two welfare objectives: a Rawlsian (Egalitarian) objective that minimizes the worst-off group’s disutility, and a Utilitarian objective that minimizes the sum of group disutilities (Zhang et al., 14 Aug 2025).

1. Conceptual lineage and motivation

The immediate intellectual background for welfare-centric clustering is the socially fair clustering literature. "Socially Fair k-Means Clustering" formalized the claim that standard Lloyd-style $k$-means can produce systematically unequal subgroup outcomes even when the global average squared distance is low. On the Adult dataset, Lloyd’s algorithm produced up to $15\%$ higher average clustering cost for females than males; for racial groups on the same dataset, the average cost for Asian-Pac-Islander individuals could be up to $4$ times worse than for white individuals; and on the Credit dataset, lower-educated individuals also experienced higher cost. That paper therefore defined fairness through the average clustering cost borne by each group and proposed Fair-Lloyd, a modification of Lloyd’s heuristic that minimizes the maximum average group cost rather than the global average alone (Ghadiri et al., 2020).

This maximin perspective was subsequently generalized beyond $k$-means. "Socially Fair Center-based and Linear Subspace Clustering" defined socially fair center-based clustering by the objective

$$\operatorname{fcost}(C,X)=\max_{j\in[\ell]}\operatorname{cost}(C,X_j),$$

where $X_1,\dots,X_\ell$ are disjoint sensitive groups, and extended the same welfare-oriented worst-group criterion to linear subspace clustering. That work also emphasized that the relevant harm is not cluster composition per se but unequal clustering error, and it provided a unified coreset-based framework for both center-based and subspace settings (Gorantla et al., 2022).

Welfare-centric clustering departs from both of these lines by arguing that neither proportional representation nor equalized distance cost is by itself a complete welfare model. The central motivation in (Zhang et al., 14 Aug 2025) is that representation-based fairness can force harmful mixing when geometric separation makes mixing expensive, whereas cost-based fairness can ignore cluster composition entirely. Fair Clustering under a Bounded Cost (FCBC) is criticized for constraining total clustering cost and then maximizing proportional mixing while still reasoning in terms of clustering cost rather than group welfare. The welfare-centric formulation therefore treats distance and representation as joint determinants of group utility, rather than optimizing one while constraining the other (Zhang et al., 14 Aug 2025).

2. Formal model of group disutility

The welfare-centric model considers a point set $\Points$ of size $n$, a group-label map $\chi:\Points\to\Colors$, and group-specific subsets $\Points^h$ of size $15\%$0. The global proportion of group $15\%$1 is

$15\%$2

A clustering solution is a pair $15\%$3, where $15\%$4 is the set of centers and $15\%$5 assigns points to centers. For standard $15\%$6-median and $15\%$7-means, the unconstrained baseline objective is

$15\%$8

with $15\%$9 for $4$0-median and $4$1 for $4$2-means (Zhang et al., 14 Aug 2025).

Welfare-centric clustering introduces a group-specific distance term and a group-specific proportional-violation term. For group $4$3, the distance component is

$4$4

If $4$5 is the cluster associated with center $4$6, and $4$7, the violation of proportional representation for group $4$8 in cluster $4$9 is

$k$0

where $k$1 and $k$2 are the permitted upper and lower slack parameters for group $k$3. If the realized cluster proportion lies within

$k$4

the violation is $k$5; otherwise it is the extent of overrepresentation or underrepresentation beyond the allowed interval (Zhang et al., 14 Aug 2025).

The total proportional violation for group $k$6 is then

$k$7

The factor $k$8 weights proportional mismatch by cluster size, so the same proportion error contributes more heavily in a larger cluster. The average disutility of group $k$9 is defined by combining the distance and proportional-violation terms: $$\operatorname{fcost}(C,X)=\max_{j\in[\ell]}\operatorname{cost}(C,X_j),$$0 where $$\operatorname{fcost}(C,X)=\max_{j\in[\ell]}\operatorname{cost}(C,X_j),$$1. When $$\operatorname{fcost}(C,X)=\max_{j\in[\ell]}\operatorname{cost}(C,X_j),$$2, only distance matters; when $$\operatorname{fcost}(C,X)=\max_{j\in[\ell]}\operatorname{cost}(C,X_j),$$3, only proportional violation matters; and intermediate values encode a direct trade-off between geometric quality and representation quality (Zhang et al., 14 Aug 2025).

3. Welfare objectives and their relation to prior fairness notions

The paper formulates two welfare objectives drawn from economic welfare theory. The Rawlsian, or Egalitarian, objective is

$$\operatorname{fcost}(C,X)=\max_{j\in[\ell]}\operatorname{cost}(C,X_j),$$4

and the optimization problem is to minimize the maximum group disutility. This protects the worst-off group by construction. The Utilitarian objective is

$$\operatorname{fcost}(C,X)=\max_{j\in[\ell]}\operatorname{cost}(C,X_j),$$5

and the optimization problem is to minimize the total disutility across groups (Zhang et al., 14 Aug 2025).

These objectives connect cleanly to earlier clustering formulations. If $$\operatorname{fcost}(C,X)=\max_{j\in[\ell]}\operatorname{cost}(C,X_j),$$6, the Rawlsian objective becomes exactly socially fair clustering: $$\operatorname{fcost}(C,X)=\max_{j\in[\ell]}\operatorname{cost}(C,X_j),$$7 If $$\operatorname{fcost}(C,X)=\max_{j\in[\ell]}\operatorname{cost}(C,X_j),$$8, the Utilitarian objective becomes a weighted clustering problem with point weights

$$\operatorname{fcost}(C,X)=\max_{j\in[\ell]}\operatorname{cost}(C,X_j),$$9

This reduction is algorithmically important because it allows welfare-centric methods to inherit center-selection procedures from socially fair clustering and weighted clustering, respectively (Zhang et al., 14 Aug 2025).

The conceptual distinction from previous fair-clustering notions is explicit. Representation-based fairness seeks cluster compositions that mirror global group proportions; socially fair clustering seeks parity or protection in group-specific clustering cost; FCBC constrains clustering cost and then maximizes proportional mixing. Welfare-centric clustering instead directly optimizes a welfare model in which distance and representation are both arguments of group disutility. The motivating example in (Zhang et al., 14 Aug 2025) is a dataset with two “top” groups far from the remaining points: strict proportional mixing can force those groups into expensive assignments, whereas a welfare-based objective can preserve separation when mixing is geometrically costly. Conversely, an equal-cost solution can still leave highly imbalanced cluster compositions, because cost-based fairness by itself does not encode any preference over representation (Zhang et al., 14 Aug 2025).

4. Algorithmic framework, relaxations, and guarantees

The algorithmic structure in (Zhang et al., 14 Aug 2025) is two-stage. First, centers are selected by an approximation algorithm suited to the limiting distance-only case. Second, with centers fixed, a linear programming relaxation is solved for the assignment, followed by a flow-based rounding procedure. The paper proves that even with fixed centers, finding the optimal assignment is NP-hard for both Rawlsian and Utilitarian welfare; the reduction is from Exact Cover by 3-Sets (Zhang et al., 14 Aug 2025).

For the Rawlsian objective, the first-stage center set $X_1,\dots,X_\ell$0 is produced by an $X_1,\dots,X_\ell$1-approximation algorithm for socially fair clustering. The assignment LP minimizes a slack variable $X_1,\dots,X_\ell$2 subject to the requirement that $X_1,\dots,X_\ell$3 upper-bound every group’s disutility: $X_1,\dots,X_\ell$4 subject to

$X_1,\dots,X_\ell$5

together with constraints

$X_1,\dots,X_\ell$6

$X_1,\dots,X_\ell$7

$X_1,\dots,X_\ell$8

and

$X_1,\dots,X_\ell$9

Here $\Points$0 is the fractional assignment of point $\Points$1 to center $\Points$2, while $\Points$3, $\Points$4, and $\Points$5 encode underrepresentation, overrepresentation, and effective proportional-violation penalties (Zhang et al., 14 Aug 2025).

The Rawlsian rounding step is not a standard one-shot min-cost flow. Because the objective is a maximum over groups rather than a sum, the paper constructs separate flow networks per color. The rounding guarantees that, for every group $\Points$6, the integral solution does not increase that group’s distance term relative to the fractional solution, and the group counts in every cluster differ from the fractional values by at most $\Points$7. If the first-stage center routine is an $\Points$8-approximation, the final Rawlsian value is bounded by

$\Points$9

For $n$0, this specializes to $n$1 times optimum plus the additive term; for $n$2, it becomes $n$3 times optimum plus the additive term (Zhang et al., 14 Aug 2025).

For the Utilitarian objective, the first-stage centers are selected by an $n$4-approximation algorithm for weighted clustering with weights $n$5. The LP objective directly minimizes the sum of group disutilities rather than introducing a slack variable $n$6. The rounding procedure is closer to standard fair-clustering flow rounding but modifies arc costs to

$n$7

and preserves both per-group cluster counts and total cluster sizes to within $n$8. The resulting guarantee is

$n$9

Because both algorithms solve an LP followed by polynomially many flow computations, the overall procedures are polynomial-time despite the underlying NP-hardness of the assignment problem (Zhang et al., 14 Aug 2025).

5. Empirical evaluation and observed behavior

The empirical study in (Zhang et al., 14 Aug 2025) evaluates the Rawlsian and Utilitarian algorithms on several UCI datasets. The two-group experiments use Adult with $\chi:\Points\to\Colors$0 points, $\chi:\Points\to\Colors$1 dimensions, and gender as the group label; CreditCard with $\chi:\Points\to\Colors$2 points, $\chi:\Points\to\Colors$3 dimensions, and marital status as the group label; and a $\chi:\Points\to\Colors$4-point sample of Census1990 with $\chi:\Points\to\Colors$5 dimensions and gender as the group label. A multi-group experiment uses Bank with $\chi:\Points\to\Colors$6 points, $\chi:\Points\to\Colors$7 dimensions, marital status as the group label, and $\chi:\Points\to\Colors$8. The feature choices follow prior fair-clustering work: Adult uses age, final-weight, education-num, capital-gain, and hours-per-week; CreditCard and Census1990 use all attributes except the group label; Bank uses age, balance, and duration-of-account (Zhang et al., 14 Aug 2025).

The evaluation varies $\chi:\Points\to\Colors$9 from $\Points^h$0 to $\Points^h$1, $\Points^h$2 over $\Points^h$3, and slack parameters according to $\Points^h$4 with $\Points^h$5. For FCBC baselines, the clustering-cost upper bound is set to $\Points^h$6 the vanilla $\Points^h$7-means cost. The Rawlsian comparisons are against vanilla $\Points^h$8-means, socially fair $\Points^h$9-means, and FCBC-Rawl; the Utilitarian comparisons are against vanilla $15\%$00-means, weighted $15\%$01-means with weights $15\%$02, and FCBC-Util. Performance is assessed using the paper’s own welfare objectives, namely the final Rawlsian value $15\%$03 and Utilitarian value $15\%$04 (Zhang et al., 14 Aug 2025).

The main findings are that the proposed Rawlsian algorithm generally dominates the baselines on the Rawlsian objective, and the proposed Utilitarian algorithm generally dominates the baselines on the Utilitarian objective. The methods are especially strong relative to FCBC variants, perform particularly well at $15\%$05 on Adult, CreditCard, and Census1990, remain competitive or better across the full range $15\%$06, and in the multi-group Bank dataset the Utilitarian method continues to outperform FCBC-Util. The appendix further reports that the LP-to-integer rounding gap is empirically very small, never exceeding about $15\%$07, which is consistent with the additive $15\%$08 theory. On Census1990 subsamples from $15\%$09k to $15\%$10k points, runtime is roughly $15\%$11–$15\%$12 seconds depending on dataset size and the welfare objective (Zhang et al., 14 Aug 2025).

6. Interpretation, related applications, and limitations

A central interpretive point is that welfare-centric clustering should not be conflated with earlier socially fair clustering, even though the Rawlsian objective reduces to socially fair clustering when $15\%$13. Socially fair clustering protects the worst-served group in terms of average clustering loss, and the literature provides both algorithmic and empirical support for that objective: Fair-Lloyd equalizes or nearly equalizes two-group costs on Adult and Credit, reduces multi-group disparities on Adult, incurs only about $15\%$14 runtime overhead on Adult and Credit and about $15\%$15 on LFW, and increases the ordinary $15\%$16-means cost by at most $15\%$17 on LFW, $15\%$18 on Adult, and $15\%$19 on Credit (Ghadiri et al., 2020). Welfare-centric clustering absorbs that worst-group perspective but adds proportional representation as an explicit argument of group disutility, thereby changing both the normative interpretation and the optimization landscape (Zhang et al., 14 Aug 2025).

The same distinction is visible in the broader socially fair clustering framework. In center-based and linear subspace settings, socially fair clustering defines fairness through worst-group cost and supports this with groupwise coresets: if $15\%$20 is a strong $15\%$21-coreset for group $15\%$22, then $15\%$23 is a strong $15\%$24-coreset for the socially fair objective. That framework yields a $15\%$25-approximation for socially fair $15\%$26-clustering and a $15\%$27-approximation for socially fair $15\%$28 linear subspace clustering, where $15\%$29 (Gorantla et al., 2022). A plausible implication is that welfare-centric clustering can be understood as a next step in the same trajectory: moving from worst-group protection on a single geometric loss to explicit welfare modeling over multiple group-level desiderata.

A related, but methodologically distinct, welfare-oriented use of clustering appears in child welfare data analysis. "Mutual Information Scoring: Increasing Interpretability in Categorical Clustering Tasks with Applications to Child Welfare Data" does not optimize a formal welfare objective over groups; instead, it uses clustering as an exploratory tool for interrogating administrative foster-care data. The proposed Mutual Information Scoring method recursively splits high-dimensional categorical data by selecting the attribute with the highest normalized average mutual information, then summarizes clusters by KL-divergence from the global population. Missing values are encoded as a separate category “?” rather than imputed, allowing the method to surface non-random missingness and state- or placement-related structure. The paper explicitly frames the approach as prescriptive and exploratory rather than predictive, and warns against using clustering to automate or replace human judgment in high-stakes child welfare decisions (Sankhe et al., 2022). This suggests that “welfare-centric” clustering in the current literature has at least two senses: direct optimization of group welfare, and clustering as a welfare-oriented analytic lens for revealing systemic patterns.

Several limitations remain explicit in the literature. Welfare-centric clustering assumes known group membership and explicit parameterization of the trade-off between distance and representation through $15\%$30, $15\%$31, and $15\%$32 (Zhang et al., 14 Aug 2025). Socially fair clustering assumes that fairness-relevant groups are fixed in advance and measures harm through average squared distance or related geometric cost, which does not capture all possible notions of disadvantage (Ghadiri et al., 2020). In the multi-group setting, even socially fair optimization becomes less structured than the two-group case; and in welfare-centric clustering, the assignment problem remains NP-hard even when centers are fixed (Zhang et al., 14 Aug 2025). The field therefore presents welfare-centric clustering not as a single closed problem, but as a family of objective-driven formulations that make explicit which group-level harms and benefits are being optimized.