A Technique for Obtaining True Approximations for $k$-Center with Covering Constraints

TL;DR

This paper introduces a novel technique for the $k$-Center problem with covering constraints, achieving true constant-factor approximations for certain fairness variants, resolving open questions in the field.

Contribution

The authors develop a new approach that yields true approximations for fair $k$-Center variants, including Colorful and Fair Robust models, which previously only had pseudo-approximations.

Findings

Achieves a 4-approximation for Colorful $k$-Center with few colors.

Provides a 4-approximation for Fair Robust $k$-Center.

Shows no finite approximation exists for unbounded colors unless P=NP.

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 $\mathrm{P} \neq \mathrm{NP}$. 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.