社会公平聚类：参数化近似与局部搜索
Socially Fair Clustering: Parameterized Approximation and Local Search
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中文总结 AI 辅助
针对社会公平聚类及其扩展问题，提出首个以组数为参数的常数因子FPT近似算法，分析简单局部搜索的近似比，还设计了非统一开设成本的设施选址变体近似算法。
中文摘要 AI 辅助
我们研究了由Abbasi、Bhaskara和Venkatasubramanian（2021）以及Ghadiri、Samadi和Vempala（2021）提出的社会公平聚类（Socially Fair Clustering）问题，及其扩展形式$(p,q)$-社会公平聚类问题。该问题是$k$-中位数和$k$-均值在数据点被划分为$\text{ℓ}$个组场景下的推广，目标是找到一个同时对所有组都表现良好的公平聚类。我们针对该问题提出了多种算法。\n对于$\text{ℓ}_p$-社会公平聚类，我们给出了首个以组数$\text{ℓ}$为参数的常数因子FPT（固定参数可处理）近似算法，解决了Ghadiri、Singh和Vempala（2022）提出的开放问题。我们的核心技术是受局部搜索启发、能在参数化时间内关闭额外中心的新算法。\n随后我们研究了更通用的$(p,q)$-社会公平聚类问题。Chlamtáč、Makarychev和Vakilian（2022）提出的该问题现有算法近似效果很好，但结构复杂、运行速度慢且难以实现。我们分析了一种简单局部搜索算法的性能，证明其在最坏情况下可达到$O(q)$的近似比。\n最后，我们为该问题的设施选址变体设计了近似算法，该变体中设施（中心）的数量不预先固定，且开设每个设施都会产生开设成本。与以往研究不同，我们不假设所有组的设施开设成本相同。
英文摘要
We study the Socially Fair Clustering problem introduced by Abbasi, Bhaskara, and Venkatasubramanian (2021) and Ghadiri, Samadi, and Vempala (2021), along with its extension, the $(p,q)$-Socially Fair Clustering problem. This problem generalizes $k$-medians and $k$-means to settings where data points are partitioned into $\ell$ groups, and the goal is to find a fair clustering that is simultaneously good for all groups. We present several algorithms for this problem. For $\ell_p$-Socially Fair Clustering, we give the first constant-factor FPT-approximation parameterized by the number of groups $\ell$, resolving the open question raised by Ghadiri, Singh, and Vempala (2022). Our main ingredient is a new algorithm for closing additional centers in parameterized time inspired by local search. We then turn to the more general $(p,q)$-Socially Fair Clustering problem. The known algorithm for this problem, proposed by Chlamtáč, Makarychev, and Vakilian (2022) achieves a very good approximation but is complex, slow and difficult to implement. We analyze the performance of a simple local search algorithm and show that it provides an $O(q)$ approximation in the worst case. Finally, we design approximation algorithms for the facility location variant of the problem, where the number of facilities (centers) is not fixed in advance, and opening each facility incurs an opening cost. Unlike in previous work, we do not assume these opening costs are the same for all groups.
发表机构
- University of Michigan（密歇根大学）
- Toyota Technological Institute at Chicago (TTIC)（芝加哥丰田技术研究所）
机构由 AI 辅助整理，请以论文原文为准。
