In this paper, we study the individual preference (IP) stability, which is an notion capturing individual fairness and stability in clustering. Within this setting, a clustering is $\alpha$-IP stable when each data point's average distance to its cluster is no more than $\alpha$ times its average distance to any other cluster. In this paper, we study the natural local search algorithm for IP stable clustering. Our analysis confirms a $O(\log n)$-IP stability guarantee for this algorithm, where $n$ denotes the number of points in the input. Furthermore, by refining the local search approach, we show it runs in an almost linear time, $\tilde{O}(nk)$.

翻译：本文研究个体偏好（IP）稳定性，这是一种在聚类中同时体现个体公平性与稳定性的概念。在该设定下，当每个数据点到其所属簇的平均距离不超过其到其他任意簇的平均距离的$\alpha$倍时，聚类结果称为$\alpha$-IP稳定的。本文针对IP稳定聚类中的自然局部搜索算法展开研究。我们的分析证实该算法具有$O(\log n)$-IP稳定性保证，其中$n$表示输入数据点的数量。此外，通过对局部搜索方法进行优化，我们证明该算法能以近乎线性的时间$\tilde{O}(nk)$运行。