  The remarkable attention which fair clustering has received in the last few
years has resulted in a significant number of different notions of fairness.
Despite the fact that these notions are well-justified, they are often
motivated and studied in a disjoint manner where one fairness desideratum is
considered exclusively in isolation from the others. This leaves the
understanding of the relations between different fairness notions as an
important open problem in fair clustering. In this paper, we take the first
step in this direction. Specifically, we consider the two most prominent
demographic representation fairness notions in clustering: (1) Group Fairness
(GF), where the different demographic groups are supposed to have close to
population-level representation in each cluster and (2) Diversity in Center
Selection (DS), where the selected centers are supposed to have close to
population-level representation of each group. We show that given a constant
approximation algorithm for one constraint (GF or DS only) we can obtain a
constant approximation solution that satisfies both constraints simultaneously.
Interestingly, we prove that any given solution that satisfies the GF
constraint can always be post-processed at a bounded degradation to the
clustering cost to additionally satisfy the DS constraint while the reverse is
not true. Furthermore, we show that both GF and DS are incompatible (having an
empty feasibility set in the worst case) with a collection of other
distance-based fairness notions. Finally, we carry experiments to validate our
theoretical findings.
