覆盖约束下$k$ -中心的真逼近技术
A Technique for Obtaining True Approximations for $k$-Center with Covering Constraints
摘要：
最近，人们对将公平性方面纳入经典聚类问题的兴趣激增。在这种精神下，$k$ -中心问题的两个最近引入的变体是由Bandyapadhyay、Inamdar、Pai和Varadarajan引入的彩色$k$ -中心问题，以及由Harris、Pensyl、Srinivasan和Trinh引入的彩票模型，例如公平稳健$k$ -中心问题。为了解决公平性方面，与传统的$k$ -中心相比，这些模型包括附加的覆盖约束。这些模型的先验近似结果需要放松一些通常的硬约束，如要打开的中心的数量或所涉及的覆盖约束，因此，仅获得常数因子的伪近似。在本文中，我们介绍了一种新的方法来处理这样的覆盖约束，导致（真）近似，包括$4$ -近似彩色$k$ -中心与恒定的许多颜色-解决了Bandyapadhyay，Inamdar，Pai和Varadarajan提出的一个开放的问题-和$4$ -近似公平鲁棒$k$ -中心，对于这一点，（真正的）常数因子近似的存在也是开放的。我们补充我们的结果表明，如果一个允许无限数量的颜色，然后彩色$k$ -中心承认没有近似算法有限近似保证，假设$\mathrm{P} \neq \mathrm{NP}$。此外，在指数时间假设下，如果颜色数量的增长速度超过基集大小的对数，则问题是不可近似的。
英文摘要：
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.
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