Clustering is a fundamental problem in unsupervised machine learning, and fair variants of it have recently received significant attention due to its societal implications. In this work we introduce a novel definition of individual fairness for clustering problems. Specifically, in our model, each point $j$ has a set of other points $\mathcal{S}_j$ that it perceives as similar to itself, and it feels that it is fairly treated if the quality of service it receives in the solution is $\alpha$-close (in a multiplicative sense, for a given $\alpha \geq 1$) to that of the points in $\mathcal{S}_j$. We begin our study by answering questions regarding the structure of the problem, namely for what values of $\alpha$ the problem is well-defined, and what the behavior of the \emph{Price of Fairness (PoF)} for it is. For the well-defined region of $\alpha$, we provide efficient and easily-implementable approximation algorithms for the $k$-center objective, which in certain cases enjoy bounded-PoF guarantees. We finally complement our analysis by an extensive suite of experiments that validates the effectiveness of our theoretical results.

翻译：集成是不受监督的机器学习中的一个根本问题, 其公平的变种最近因其社会影响而受到极大关注。 在这项工作中, 我们引入了个人对集群问题公平性的新定义。 具体地说, 在我们的模型中, 每点美元都有一系列其他的点数 $mathcal{S ⁇ j$, 它自认为与自己相似, 并且它觉得如果它在解决方案中获得的服务质量质量为$alpha$- close( 从多种意义上讲, 给的$\alpha\ ge $1美元) 与$\ mathcal{S ⁇ j$的点的点数相比, 它得到了公平对待。 我们从研究开始, 我们从回答关于问题结构的问题开始, 即 $\ alpha$ 的值是多少, 问题本身认为与自己相似, 以及 它的行为是什么样的。 如果它在解决方案中得到的服务质量是 $alpha- closeal, 我们为$- center目标提供了高效和易于执行的近似的近算算算算算算算算算法, 我们最终享受了理论上的验证结果。