[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84810-en":3,"doc-seo-84810-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84810,2336464648322,"Aria","https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488",8,"Research & Report","Exact ratio preservation via outliers for fair k-center clustering","Exact ratio preservation via outliers for fair k-center clustering studies k-center clustering with demographic fairness constraints, where points are partitioned into groups and each cluster should reflect a target group proportion. Prior approaches assume the input already satisfies proportions or rely on relaxed fairness. The work introduces a model combining the fair clustering framework of Chierichetti et al. with outliers, enabling constant-factor approximate solutions that exactly match specified ratios for multiple group settings, including two- and m-group proportions.","arXiv :2607 .05342v2 [ cs .DS] 9 Jul 2026  \nExact ratio preservation via outliers for fair k-center clustering  \nAnna Arutyunova \\#  \nHeinrich Heine University Düsseldorf, Faculty of Mathematics and Natural Sciences, Germany Irina Fast \\#  \nHeinrich Heine University Düsseldorf, Faculty of Mathematics and Natural Sciences, Germany Annika Hennes \\# 􀀚  \nHeinrich Heine University Düsseldorf, Faculty of Mathematics and Natural Sciences, Germany Carsten Krollmann \\#  \nHeinrich Heine University Düsseldorf, Faculty of Mathematics and Natural Sciences, Germany Daniel R. Schmidt \\# 􀀚  \nHeinrich Heine University Düsseldorf, Faculty of Mathematics and Natural Sciences, Germany Melanie Schmidt \\# 􀀚  \nHeinrich Heine University Düsseldorf, Faculty of Mathematics and Natural Sciences, Germany  \n~~ Abstract ~~  \nWe study the k-center clustering problem under demographic fairness constraints, where the point set is partitioned into groups, and the aim is to compute clusters that exhibit a given group proportion. Previous work in this direction assumes that the entire point set already respects the desired proportions or uses relaxed notions of fairness.  \nIn this work, we propose a model that facilitates the creation of clusters that exactly match given target ratios, even when the input point set does not. We combine the well-known fair clustering model initiated by Chierichetti, Kumar, Lattanzi, and Vassilvitskii [9] with the notion of outliers to obtain a practical combinatorial framework that provides constant-factor approximate solutions for all proportion settings from 1 ∶ 1 for two groups to t1 ∶ t2 ∶ . . . ∶ tm for m ≥ 2 groups, where t1 , . . . , tmare integers.  \nWe implement and evaluate our algorithms, compare different variants, and provide evidence of the practicability of this approach.  \n2012 ACM Subject Classification Theory of computation → Facility location and clustering; Theory of computation → Approximation algorithms analysis  \nKeywords and phrases Fairness, k-center, approximation algorithms  \nSupplementary Material  \nSoftware (Source Code): [https://github.com/algo-hhu/fair-k-center-via-outliers/](https://github.com/algo-hhu/fair-k-center-via-outliers/)  \nFunding Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Project 456558332 through the Emmy Noether Programme  \nA. Arutyunova, I. Fast, A. Hennes, C. Krollmann, D. Schmidt and M. Schmidt 1  \n 1  Introduction  \nFair clustering is a very active line of research in clustering algorithm design, introduced by Chierichetti, Kumar, Lattanzi, and Vassilvitskii [9] in 2017 . The key idea is that, when applying k-clustering, the user may specify a protected attribute such as gender, race, or another demographic variable to which the algorithm should pay particular attention. The goal is to ensure group-level fairness: each cluster shall contain the same proportion of attribute values (i. e., demographic groups) as observed in the overall dataset. That is, the composition of every cluster shall match the global distribution of the protected attribute.  \nEnforcing such fairness can be valuable in a variety of real-world settings, as it helps ensure diversity within each group. This is why the model is called fair: It makes sure that every group is adequately represented in each cluster. Fairness is a desired or necessary condition in many applications, e. g. , representation of people with protected characteristics in committees, creating geographic zones with demographic constraints (e.g., schools [21]), allocation of scarce resources (e.g., access to childcare, or charging times for electric vehicles [24]), or preventing dominance of a single actor in ads [1] .  \nTo further explain the model, let us consider a simple base case (the general case is defined in Section 1.2): Given is a set of points P , a metric d ∶ P × P → R≥0, the desired number of centers k, and a mapping γ ∶ P → {blue, red} that satisfies ∣γ−1 (blue)∣ = ∣γ−1 (red)∣, i. e. , the i","cbCaitxxHIg9trr9","https://ap.wps.com/l/cbCaitxxHIg9trr9","pdf",1061209,1,49,"English","en",105,"# Introduction\n## Fair clustering and the fair k-center model\n## Fairlet decomposition and constant-factor ideas\n## Motivation for relaxing exact proportionality","[{\"question\":\"What fairness objective does the document target in k-center clustering?\",\"answer\":\"It targets group-level demographic fairness: the proportion of each protected group in every cluster matches the desired target proportion (e.g., equal red/blue counts in the two-color case).\"},{\"question\":\"Why is exact ratio matching challenging in real-world inputs?\",\"answer\":\"When the available group counts in the input do not align with the target ratio, the only subset that could match may be the entire dataset, making strict fairlet decompositions impractical or degenerate.\"},{\"question\":\"How do outliers help achieve exact ratio preservation?\",\"answer\":\"The approach integrates outliers into the fair clustering framework, producing a combinatorial model that yields constant-factor approximate solutions across all target proportion settings, enabling exact ratio matching even when the full input does 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fairness objective does the document target in k-center clustering?","Question",{"text":75,"@type":76},"It targets group-level demographic fairness: the proportion of each protected group in every cluster matches the desired target proportion (e.g., equal red/blue counts in the two-color case).","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why is exact ratio matching challenging in real-world inputs?",{"text":80,"@type":76},"When the available group counts in the input do not align with the target ratio, the only subset that could match may be the entire dataset, making strict fairlet decompositions impractical or degenerate.",{"name":82,"@type":73,"acceptedAnswer":83},"How do outliers help achieve exact ratio preservation?",{"text":84,"@type":76},"The approach integrates outliers into the fair clustering framework, producing a combinatorial model that yields constant-factor approximate solutions across all target proportion settings, enabling exact ratio matching even when the 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