Title: Unifying Proportional Fairness in Centroid and Non-Centroid Clustering
Venue: NeurIPS 2025
Accepted: yes
Keywords: Proportional Fairness

Clustering

Algorithmic Fairness
Abstract: Proportional fairness criteria inspired by democratic ideals of proportional representation have received growing attention in the clustering literature. Prior work has investigated them in two separate paradigms. Chen et al. [ICML 2019] study _centroid clustering_, in which each data point's loss is determined by its distance to a representative point (centroid) chosen in its cluster. Caragiannis et al. [NeurIPS 2024] study _non-centroid clustering_, in which each data point's loss is determined by its maximum distance to any other data point in its cluster. 
  
We generalize both paradigms to introduce _semi-centroid clustering_, in which each data point's loss is a combination of its centroid and non-centroid losses, and study two proportional fairness criteria---the core and, its relaxation, fully justified representation (FJR). Our main result is a novel algorithm which achieves a constant approximation to the core, in polynomial time, even when the distance metrics used for centroid and non-centroid loss measurements are different. We also derive improved results for more restricted loss functions and the weaker FJR criterion, and establish lower bounds in each case.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies proportional clustering. In particular, it studies a setting generalizing both centroid and non-centroid clustering. That is, the goal is to partition a set of points (like in non-centroid clustering) and assign each partition to a center (like in centroid clustering) at the same time, with the loss of each agent being a combination of its centroid and non-centroid loss. The objective studied in the paper is proportional fairness. That is, ideally, with n points and k cluster centers approximately n/k points should deserve their own partition and center. The paper introduces several definitions of proportional fairness (depending on the loss function of the points and the strength of the deviating coalitions) and gives algorithms and lower bounds for achieving these proportional fairness definitions.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: Proportional fairness is an interesting topic and this paper makes a novel contribution to it that should be of interest to the researchers working on proportional fair clustering. In fact, I have  asked myself something similar akin to Theorem 1 while reading through the paper of Caragiannis et al. and am happy that there is a nice simple (albeit negative) solution to this. I also find the study of combining centroid and non-centroid clustering to be somewhat interesting and novel. The algorithms presented throughout the paper seem are for the most part non-trivial and clearly presented, and I think the authors largely do a good job at making the paper readable. The results also seem technically interesting, and I think the two-phase Algorithm 1 is a nice twist on the usual Greedy Capture based algorithms.\n\nWeaknesses: \n\nI think one obvious weakness of the paper is the setting. In particular (while seemingly novel), I find the study of both centroid and non-centroid clustering at the same time to be slightly unnatural, and I think it could have been better motivated in the paper. Are you aware of any other papers studying this setting or a related setting where one might need both centroid and non-centroid clustering?\n\nSecondly, one thing I found a bit unnatural, is that in the setting in which N = M, for a given cluster (C, x) the cluster center x must not necessarily be a part of C (and this is also not enforced by the algorithms as far as I can see). Do the results still hold if this is enforced or might this not even be desirable?\n\nThe presentation of the paper could be improved at times. In particular, I think the bounds presented in Lemma 4, Theorem 5, Theorem 6 should be explained. For further things, see detailed comments later. \n\nFinally, while I usually think that a theory paper does not need experiments, I think they would be nice. In fact, the paper even has experiments and I would have loved to see them in the main text. In particular, I would have been interested to see whether the worst case presented in Theorem 1 actually happens in real world instances.\n\nDetailed Comments:\n-The equivalence between the two forms of k-means presented at the start of the paper do not hold for the setting here, in which there is separate candidate set.\n- “which is a principle at the heart of our democratic institutions.“ I am not a fan of this phrasing. Proportionality is not even at the heart of American institutions. \n-I find Table 1 to be quite unreadable without context. In particular, I think that every variable in a table should be introduced to make the table understandable.\n-”approximation to the core” without context, it is not known what kind of “approximation” this is supposed to be\n- For stylistic purposes, the order of the losses should be the same in lines 129 and 136\n- I think sorted references are easier to access for the reader. Also, you currently have abbreviated first names for some, and spelled out for others, papers 13 and 17 are no longer forthcoming, and paper 28 has a conference version.\n\nQUESTIONS:\nI would be interested to hear answers to the two questions I had in my weaknesses section. \n\n1. Are you aware of any other papers studying this setting or a related setting where one might need both centroid and non-centroid clustering?\n2. In the setting in which N = M, for a given cluster (C, x) the cluster center x must not necessarily be a part of C (and this is also not enforced by the algorithms as far as I can see). Do the results still hold if this is enforced or might this not even be desirable?\n\nWith sufficient answer to these questions (in particular the first question) I am willing to upgrade to an accept.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nMotivated by the fact that the main algorithms for proportional non-centroid and centroid clustering (called Greedy Capture) look almost the same,\nthe authors consider a clustering variant where one simultaneously computes a non-centroid and a centroid clustering, for possibly separate metrics.\nThe resulting problems partitions the given points into k clusters and maps a center point to each of these clusters.\nThe goal is to find such a clustering that is (approximately) proportional simultaneously for the centroid-based and the non-centroid based loss function, i.e.,\nthe distance to the center and the maximum distance to any other point in the same cluster.\nThe authors present a modification of the Greedy Capture algorithm, where one first builds a tentative cluster and then allows some agents to swap clusters and makes use of a subroutine that is NP-hard to compute, but efficiently approximable.\nAssuming access to optimal solutions to the subroutine, the authors show that there always exists a clustering in the 3-core,\nand using the approximate subroutine, the authors show that one can efficiently compute a clustering in the (3+2sqrt(3))-core.\n\nThe authors further show some results for the case that centroid and non-centroid clustering use the same metric and also consider a relaxation of the core notion called FJR.\nThese results are based mainly on small variations of existing algorithms.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n- From a theoretical perspective, the authors approach the intriguing question why almost the same algorithm works for two quite different variants of clustering (albeit with similar fairness notions) and find a satisfying answer: For optimizing both notions at the same time, one needs to deviate a bit further from the Greedy Capture approach. The practical motivation via federated learning also seems plausible in the sense that, if I would rather deviate to another model, then I also need to find a group of peers who would join me in the deviation.\n- The paper presents itself very well, the proofs are clearly written and easy to verify.\n\nWeaknesses\n- While the theoretical motivation is very strong, the practical motivation is a bit faltering to me. I do believe that the federated learning motivation could somehow be interesting, and I like the idea of having the centroids represent a model. I do believe that it would help add more details to this motivation.\n- Overall, the technical depth of the work is not too high. The new Algorithm 1 is a nice modification from Greedy Capture, but still lends many elements from the algorithms by Chen et al. and Caragiannis et al., such as the most cohesive cluster subroutine.\n- The impact of the work is not too high. In all, the work seems a bit niche and incremental.\n\nQUESTIONS:\n- What is the best factor alpha obtainable by Algorithm 1 when only focusing only on the centroid loss or only on the non-centroid loss?\n- Suppose that you are given a non-centroid clustering in the alpha-non-centroid core. Is it possible to extend this to a semi-centroid clustering that is in the alpha*beta-core for some constant beta?\n\n- I suggest that you slightly extend on the motivation for federated learning. It seems pretty plausible, but needs to be more detailed if the greater audience should be able to follow.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper the authors explore two notions of fairness in a clustering model that lies between centroid and non-centroid clustering. The task is to partition a set of $n$ data points (agents) into $k$ clusters, each defined by a center point and a group of assigned agents. An agents loss is defined as the sum of the distance to its assigned center and the maximum distance to any other agent within the same cluster. A clustering is considered unfair if there exists a group of $\\frac{n}{k}$ agents who could form their own cluster in which each member would experience a (by some amount) lower loss.\n\nThe authors analyze two fairness notions—the 'core' and 'fully justified representation'— and provide both upper and lower bounds on their approximation. They distinguish between two settings: one where the distance towards the center and the distance to other agents are measured using different metrics and another where the two metrics are identical, up to a constant factor. For both settings, the paper presents efficient algorithms that achieve constant-factor approximations of the fairness notions.\n\nSTRENGTHS AND WEAKNESSES:\nThe authors approach a natural question: Given existing fairness results for both centroid and non-centroid clustering, what if an agent cares about both being close to the center as well as to other members of their cluster? While the application-based motivation is somewhat weak, the problem itself is interesting and bridges the gap between different approaches in the literature.\nThe paper is well written and gives a clear overview of prior related work. The results seem significant and contribute to the understanding of fairness in clustering. With most of the proofs in the appendix, the authors generally succeed in conveying the key ideas behind their results and algorithms. That said, Theorem 1 would benefit from including the example instance from the proof in the main body. For Theorem 4 and 5 (or result summary table) a figure plotting the upper and lower bounds as a function of $\\lambda$ would greatly enhance clarity and help readers to interpret the bounds and the gap between them. \n\nTypo in line 226: with respect the -> with respect to the\n\nQUESTIONS:\nWhat is the motivation for allowing two different metrics in the loss function? Why would an agent measure the distance to the center differently than the distance to the other agents?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces and investigates a generalized clustering framework called semi-centroid clustering, which unifies the two previously studied paradigms in proportionally fair clustering: centroid-based clustering (where loss is defined by distance to a cluster centroid) and non-centroid clustering (where loss depends on distances to other agents in the cluster). By simply combining the two associated cost functions under different settings, the authors initiate study on two important concepts for clustering, the core and fully justified representation (FJR). The major contribution in balancing these two clustering notions is a non-trivial polynomial-time constant approximation algorithm for the core under different loss functions for the two (simultaneous) objectives.\n\nSTRENGTHS AND WEAKNESSES:\nSTRENGTHS\n\nThis paper makes a significant theoretical contribution by bridging two previously distinct lines of research in proportionally fair clustering.\nThe dual and weighted metric loss models serve as natural and flexible interpolants between existing paradigms, readily allowing extension beyond the two main directions of clustering algorithm design.\nThe technical results are nontrivial, with the core approximation algorithm under dual metric loss is a highlight.\nIn the weighted single metric loss setting, the authors provide a nuanced analysis with a full picture of the approximation frontier as parametrized by a trade-off.\nThe authors further validate their problem definition and algorithm with experimental results, though these are entirely deferred to the appendix.\n\nWEAKNESSES\n\nMy main concern is that, while the results and framing are very interesting, the complete paper is largely contained in the appendix and suggests the work may actually be better situated for a more theory heavy conference.\nA short section in the main body summarizing key empirical findings would improve accessibility and demonstrate the algorithms’ relevance to real-world clustering scenarios.\nThe paper focuses on linear interpolation between the two cost metrics. The work does not explore alternatives to these, or discuss more expressive models\n\nThe one writing weakness I found is that the algorithm descriptions are a bit hard to parse for a non-expert in the field. If possible (with a longer page limit) more intuitive examples or figures to illustrate algorithm behavior would be beneficial to the reader.\n\n\nMINOR FIXES:\n- some inconsistency in the text with \"semi centroid\" vs \"semi-centroid\"\n- the presentation of Algorithm 1 is very cumbersome with notation\n- no figures or presentation of experimental results in main text\n\nQUESTIONS:\nHow should a practitioner choose between different values of $\\lambda$ in a real application? How sensitive are the algorithms to the choice of metric or to approximation factors in MCC computation?\n\nHow do these algorithms scale in practice / what is their runtime guarantee?\n\nHow might general loss formulations might be captured in future work?\n\nLIMITATIONS:\nWhile the paper acknowledges that simultaneous core approximation is infeasible, it would be valuable to better discuss practical implications", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-05"}