We consider the problem of covering multiple submodular constraints. Given a finite ground set
Approximation Algorithms for Socially Fair Clustering
We present an $e^{O(p)} (\log \ell) / (\log \log \ell)$approximation algorithm for socially fair clustering with the $\ell_p$objective. In this problem, we are given a set of points in a metric space. Each point belongs to one (or several) of $\ell$ groups. The goal is to find a $k$medians, $k$means, or, more generally, $\ell_p$clustering that is simultaneously good for all of the groups. More precisely, we need to find a set of $k$ centers $C$ so as to minimize the maximum over all groups $j$ of $\sum_{u \text{ in group } j} d(u, C)^p$. The socially fair clustering problem was independently proposed by Abbasi, Bhaskara, and Venkatasubramanian (2021) and Ghadiri, Samadi, and Vempala (2021). Our algorithm improves and generalizes their $O(\ell)$approximation algorithms for the problem. The natural LP relaxation for the problem has an integrality gap of $\Omega(\ell)$. In order to obtain our result, we introduce a strengthened LP relaxation and show that it has an integrality gap of $\Theta((\log \ell) / (\log \log \ell))$ for a fixed p. Additionally, we present a bicriteria approximation algorithm, which generalizes the bicriteria approximation of Abbasi et al. (2021).
 Editors:
 Belkin, Mikhail; Kpotufe, Samor
 Publication Date:
 NSFPAR ID:
 10336944
 Journal Name:
 Proceedings of the Conference on Learning Theory, PMLR
 Volume:
 134
 Page Range or eLocationID:
 32463264
 Sponsoring Org:
 National Science Foundation
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