Abstract
There has been a recent surge of interest in incorporating fairness aspects into classical clustering problems. Two recently introduced variants of the k-Center problem in this spirit are Colorful k-Center, introduced by Bandyapadhyay, Inamdar, Pai, and Varadarajan, and lottery models, such as the Fair Robust k-Center problem introduced by Harris, Pensyl, Srinivasan, and Trinh. To address fairness aspects, these models, compared to traditional k-Center, include additional covering constraints. Prior approximation results for these models require to relax some of the normally hard constraints, like the number of centers to be opened or the involved covering constraints, and therefore, only obtain constant-factor pseudo-approximations. In this paper, we introduce a new approach to deal with such covering constraints that leads to (true) approximations, including a 4-approximation for Colorful k-Center with constantly many colors—settling an open question raised by Bandyapadhyay, Inamdar, Pai, and Varadarajan—and a 4-approximation for Fair Robust k-Center, for which the existence of a (true) constant-factor approximation was also open. We complement our results by showing that if one allows an unbounded number of colors, then Colorful k-Center admits no approximation algorithm with finite approximation guarantee, assuming that mathtt {P}ne mathtt {NP}. Moreover, under the Exponential Time Hypothesis, the problem is inapproximable if the number of colors grows faster than logarithmic in the size of the ground set.
Highlights
Along with k-Median and k-Means, k-Center is one of the most fundamental and heavily studied clustering problems
We present techniques for obtaining approximations for two recent fairness-inspired generalizations of k-Center along axis (ii), namely (i) γ -Colorful k-Center, as introduced by Bandyapadhyay et al [3], and (ii) Fair Robust k-Center, a lottery model introduced by Harris et al [18]
We introduce the following generalization thereof that can be handled with our techniques, which we name Fair γ -Colorful k-Center problem (Fair γ CkC). (The Fair Robust k-Center problem, as introduced in [18], corresponds to γ = 1.)
Summary
Along with k-Median and k-Means, k-Center is one of the most fundamental and heavily studied clustering problems. A related class of models that has received significant attention assumes that the ground set is colored, but requires that the ratio between colors within each cluster is approximately the same as the global ratio between colors Such variants have been considered for kMedian, k-Means, and k-Center, e.g., see [2,4,5,12,28] and references therein. For any constant ε > 0, only a (1 − ε)-fraction of the required number of elements are covered, and element u ∈ X is covered only with probability (1 − ε) p(u) instead of p(u) It was left open in [18] whether a true approximation may exist for Fair Robust k-Center
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