We study the problem of fair k-median where each cluster is required to have a fair representation of individuals from different groups. In the fair representation k-median problem, we are given a set of points X in a metric space. Each point x ∈ X belongs to one of ℓ groups. Further, we are given fair representation parameters αj and β_j for each group j ∈ [ℓ]. We say that a k-clustering C_1, ⋅⋅⋅, C_k fairly represents all groups if the number of points from group j in cluster C_i is between α_j |C_i| and β_j |C_i| for every j ∈ [ℓ] and i ∈ [k]. The goal is to find a set of k centers and an assignment such that the clustering defined by fairly represents all groups and minimizes the ℓ_1-objective ∑_{x ∈ X} d(x, ϕ(x)).
We present an O(log k)-approximation algorithm that runs in time n^{O(ℓ)}. Note that the known algorithms for the problem either (i) violate the fairness constraints by an additive term or (ii) run in time that is exponential in both k and ℓ. We also consider an important special case of the problem where and for all j ∈ [ℓ]. For this special case, we present an O(log k)-approximation algorithm that runs in time.
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A New Notion of Individually Fair Clustering: alpha-Equitable k-Center
Clustering is a fundamental problem in unsupervised machine learning, and due to its numerous societal implications fair variants of it have recently received significant attention. In this work we introduce a novel definition of individual fairness for clustering problems. Specifically, in our model, each point j has a set of other points S(j) that it perceives as similar to itself, and it feels that it is being fairly treated if the quality of service it receives in the solution is α-close (in a multiplicative sense, for some given α ≥ 1) to that of the points in S(j). We begin
our study by answering questions regarding the combinatorial structure of the problem, namely for what values of α the problem is well-defined, and what the behavior of the Price of Fairness (PoF) for it is. For the well-defined region of α, we provide efficient and easily-implementable approximation algorithms for the k-center objective, which in certain cases also enjoy bounded-PoF guarantees. We finally complement our analysis by an extensive suite of experiments that
validates the effectiveness of our theoretical results.
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- Award ID(s):
- 1918749
- NSF-PAR ID:
- 10403947
- Date Published:
- Journal Name:
- Proc. International Conference on Artifical Intelligence and Statistics (AISTATS)
- Page Range / eLocation ID:
- 6387-6408
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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