  Clustering is a fundamental unsupervised learning problem where a dataset is
partitioned into clusters that consist of nearby points in a metric space. A
recent variant, fair clustering, associates a color with each point
representing its group membership and requires that each color has
(approximately) equal representation in each cluster to satisfy group fairness.
In this model, the cost of the clustering objective increases due to enforcing
fairness in the algorithm. The relative increase in the cost, the ''price of
fairness,'' can indeed be unbounded. Therefore, in this paper we propose to
treat an upper bound on the clustering objective as a constraint on the
clustering problem, and to maximize equality of representation subject to it.
We consider two fairness objectives: the group utilitarian objective and the
group egalitarian objective, as well as the group leximin objective which
generalizes the group egalitarian objective. We derive fundamental lower bounds
on the approximation of the utilitarian and egalitarian objectives and
introduce algorithms with provable guarantees for them. For the leximin
objective we introduce an effective heuristic algorithm. We further derive
impossibility results for other natural fairness objectives. We conclude with
experimental results on real-world datasets that demonstrate the validity of
our algorithms.
