On the Cost of Essentially Fair Clusterings

Authors Ioana O. Bercea, Martin Groß, Samir Khuller, Aounon Kumar, Clemens Rösner, Daniel R. Schmidt , Melanie Schmidt

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Author Details

Ioana O. Bercea
  • School of Electrical Engineering, Tel Aviv University, Israel
Martin Groß
  • School of Business and Economics, RWTH Aachen, Germany
Samir Khuller
  • Department of Computer Science, Northwestern University, Evanston, USA
Aounon Kumar
  • Department of Computer Science, University of Maryland, College Park, USA
Clemens Rösner
  • Institute of Computer Science, University of Bonn, Germany
Daniel R. Schmidt
  • Institute of Computer Science, University of Cologne, Germany
Melanie Schmidt
  • Institute of Computer Science, University of Bonn, Germany


The authors would like to thank Sorelle Friedler for useful discussions related to the topic of fairness. The first author’s work was done while a Ph.D. student at the University of Maryland.

Cite AsGet BibTex

Ioana O. Bercea, Martin Groß, Samir Khuller, Aounon Kumar, Clemens Rösner, Daniel R. Schmidt, and Melanie Schmidt. On the Cost of Essentially Fair Clusterings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 18:1-18:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Clustering is a fundamental tool in data mining and machine learning. It partitions points into groups (clusters) and may be used to make decisions for each point based on its group. However, this process may harm protected (minority) classes if the clustering algorithm does not adequately represent them in desirable clusters - especially if the data is already biased. At NIPS 2017, Chierichetti et al. [Flavio Chierichetti et al., 2017] proposed a model for fair clustering requiring the representation in each cluster to (approximately) preserve the global fraction of each protected class. Restricting to two protected classes, they developed both a 4-approximation for the fair k-center problem and a O(t)-approximation for the fair k-median problem, where t is a parameter for the fairness model. For multiple protected classes, the best known result is a 14-approximation for fair k-center [Clemens Rösner and Melanie Schmidt, 2018]. We extend and improve the known results. Firstly, we give a 5-approximation for the fair k-center problem with multiple protected classes. Secondly, we propose a relaxed fairness notion under which we can give bicriteria constant-factor approximations for all of the classical clustering objectives k-center, k-supplier, k-median, k-means and facility location. The latter approximations are achieved by a framework that takes an arbitrary existing unfair (integral) solution and a fair (fractional) LP solution and combines them into an essentially fair clustering with a weakly supervised rounding scheme. In this way, a fair clustering can be established belatedly, in a situation where the centers are already fixed.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Facility location and clustering
  • Theory of computation → Rounding techniques
  • Theory of computation → Unsupervised learning and clustering
  • approximation
  • clustering
  • fairness
  • LP rounding


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