  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.
