Clustering with outliers is one of the most fundamental problems in Computer Science. Given a set $X$ of $n$ points and two integers $k$ and $m$, the clustering with outliers aims to exclude $m$ points from $X$ and partition the remaining points into $k$ clusters that minimizes a certain cost function. In this paper, we give a general approach for solving clustering with outliers, which results in a fixed-parameter tractable (FPT) algorithm in $k$ and $m$, that almost matches the approximation ratio for its outlier-free counterpart. As a corollary, we obtain FPT approximation algorithms with optimal approximation ratios for $k$-Median and $k$-Means with outliers in general metrics. We also exhibit more applications of our approach to other variants of the problem that impose additional constraints on the clustering, such as fairness or matroid constraints.

翻译：含离群点的聚类问题是计算机科学中最基础的问题之一。给定一个由$n$个点构成的集合$X$以及两个整数$k$和$m$，含离群点的聚类旨在从$X$中排除$m$个点，并将剩余点划分为$k$个聚类，以最小化某个特定代价函数。本文提出了一种求解含离群点聚类问题的通用方法，该方法在参数$k$和$m$上实现了固定参数可解（FPT）算法，其近似比几乎达到无离群点情况的对应水平。作为推论，我们在一般度量空间中获得了针对含离群点的$k$-中位数和$k$-均值问题的最优近似比FPT近似算法。此外，我们还展示了该方法在其他聚类变体问题中的应用——例如引入公平性约束或拟阵约束等额外限制的聚类场景。