[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117122-en":3,"doc-seo-117122-105":30,"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":4,"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":27,"seo_description":14,"update_tm":28,"read_time":29},117122,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Approximate Group Fairness for Clustering - Abstract and Methods","Group fairness is incorporated into the centroid clustering problem, aiming to place k centers for n agents in a metric space so that coalitions cannot improve their collective cost by coordinated deviations. The work refines proportional fairness into core fairness under transferable utilities, then provides existence, hardness, and approximability results. Core requirements are relaxed along two axes—degree of distance improvement and coalition size—followed by analysis and approximation algorithms for metric spaces including the line, tree, and general metrics.","Approximate Group Fairness for Clustering  \nBo Li 1 Lijun Li 2 Ankang Sun 3 Chenhao Wang 4 * Yingfan Wang 5  \nAbstract  \nWe incorporate group fairness into the algorithmic centroid clustering problem, where k centers are to be located to serve n agents distributed in a metric space. We refine the notion of proportional fairness proposed in [Chen et al., ICML 2019] as core fairness, and k-clustering is in the core if no coalition containing at least n=k agents can strictly decrease their total distance by deviating to a new center together. Our solution concept is motivated by the situation where agents are able to coordinate and utilities are transferable. A string of existence, hardness and approximability results is provided. Particularly, we propose two dimensions to relax core requirements: one is on the degree of distance improvement, and the other is on the size of deviating coalition. For both relaxations and their combination, we study the extent to which relaxed core fairness can be satisfied in metric spaces including line, tree and general metric space, and design approximation algorithms accordingly.  \n1. Introduction  \nMotivated by various real-world machine learning algorithm deployments where the data points are real human beings who should be treated unbiasedly, fairness is increasingly concerned. Most traditional algorithms mainly focus on the efficiency or profit, and thus fail to ensure fairness for individual point or collection of points. Accordingly, the past several years have seen considerable efforts in developing fair learning algorithms (Chierichetti et al., 2017 ; Chenet al., 2019 ; Backurs et al., 2019 ; Bera et al., 2019) .  \nFollowing (Chen et al., 2019), we revisit the group fairness  \n1Department of Computing, The Hong Kong Polytechnic University, Hong Kong, China 2 School of Mathematical Sciences, Ocean University of China, Qingdao, China 3Warwick Business School, University of Warwick, United Kingdom 4University of Nebraska-Lincoln, United States 5Department of Computer Science, Duke University, United States. Correspondence to: Chenhao Wang \u003C[chenhwang4-c@my.cityu.edu.hk](chenhwang4-c@my.cityu.edu.hk)>.  \nProceedings of the 38 th International Conference on Machine Learning, PMLR 139, 2021 . Copyright 2021 by the author(s) .  \nin unsupervised learning – specifically, centroid clustering. A canonical clustering problem is described as: given a metric space X with distance measure d : X 􀀂 X ! R+ [ f0g, a multiset N 􀀒 X of n data points (in this work, each data point is an agent), a set M 􀀒 X of possible centers and a positive integer k, the task is to find a subset Y 􀀒 M of k cluster centers and assign each data point to its closest center in Y. The commonly studied objective is to make data points to be as close to their assigned centers as possible. Standard algorithms, such as k-means and kmedians, solve the clustering problem by satisfying a global criterion, where individual or group-wise happiness has not been taken into consideration. It has been noted that the globally efficient solutions are less preferred, especially when the application scenario is about public resources allocation (Conitzer et al., 2017 ; Fain et al., 2018) . We consider the following facility location problem proposed in (Chen et al., 2019 ; Micha and Shah, 2020) .  \nExample 1. Suppose the government plans to build k = 11 identical parks to serve the residents, where every resident's cost (e.g. gasoline) is proportional to the distance between her home and the closest park. There is a dense urban center with a population of 10 ; 000 and 10 small suburbs, each of which has a population of 100 . Suppose the suburbs are close to each other (e.g. 10km) compared with the distance between them and the urban center (e.g. 500km).  \nAccordingly, by k-means or k-medians algorithms, the government will build 1 park at the urban center, and 10 parks for each small suburb. It is not hard to see such a plan isnot fair: a single park i","cbCaialvY3p1IsfC","https://ap.wps.com/l/cbCaialvY3p1IsfC","pdf",828089,1,11,"English","en",105,"# Introduction\n## Motivation and fairness in clustering\n## Centroid clustering and facility location formulation\n# Core fairness and transferable utilities\n## Proportional fairness vs. core fairness\n## Blocking coalitions and definition of core\n# Relaxations and approximation results\n## Relaxing improvement degree\n## Relaxing coalition size\n## Metric spaces and corresponding approximation algorithms","[{\"question\":\"What fairness notion is proposed for centroid clustering?\",\"answer\":\"The work introduces core fairness for k-clustering under transferable utilities, refining proportional fairness to prevent any sufficiently large coalition from jointly improving its total distance by deviating to a new center.\"},{\"question\":\"How is a blocking coalition defined in the core fairness framework?\",\"answer\":\"A coalition S is blocking for a clustering Y if there exists a new center y' such that the coalition’s total distance strictly decreases when all agents deviate to y' together.\"},{\"question\":\"What aspects are relaxed to obtain approximate core fairness?\",\"answer\":\"Two relaxations are studied: allowing a limited degree of distance improvement and allowing deviations by coalitions of reduced size; the paper also examines their combined effect and the resulting satisfaction of relaxed core fairness in different metric spaces.\"}]","Approximate Group Fairness for Clustering - Abstract and Methods | PDF",1785674003,28,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"approximate-group-fairness-for-clustering-abstract-and-methods","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/approximate-group-fairness-for-clustering-abstract-and-methods/117122/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What fairness notion is proposed for centroid clustering?","Question",{"text":75,"@type":76},"The work introduces core fairness for k-clustering under transferable utilities, refining proportional fairness to prevent any sufficiently large coalition from jointly improving its total distance by deviating to a new center.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is a blocking coalition defined in the core fairness framework?",{"text":80,"@type":76},"A coalition S is blocking for a clustering Y if there exists a new center y' such that the coalition’s total distance strictly decreases when all agents deviate to y' together.",{"name":82,"@type":73,"acceptedAnswer":83},"What aspects are relaxed to obtain approximate core fairness?",{"text":84,"@type":76},"Two relaxations are studied: allowing a limited degree of distance improvement and allowing deviations by coalitions of reduced size; the paper also examines their combined effect and the resulting satisfaction of relaxed core fairness in different metric spaces.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]