# Tight FPT Approximations for $k$-Median and $k$-Means

## TL;DR

This paper presents fixed-parameter tractable algorithms that achieve near-optimal approximation ratios for $k$-median and $k$-means clustering in metric spaces, and establishes hardness results indicating these ratios are essentially best possible under certain complexity assumptions.

## Contribution

The authors develop FPT algorithms with improved approximation factors for $k$-median and $k$-means, and prove matching hardness bounds under complexity conjectures.

## Key findings

- Achieved approximation ratios of (1+2/e+ε) for $k$-median and (1+8/e+ε) for $k$-means.
- Established FPT hardness results showing no better ratios are possible under certain conjectures.
- Provided insights into the complexity landscape of clustering problems in metric spaces.

## Abstract

We investigate the fine-grained complexity of approximating the classical $k$-median / $k$-means clustering problems in general metric spaces. We show how to improve the approximation factors to $(1+2/e+\varepsilon)$ and $(1+8/e+\varepsilon)$ respectively, using algorithms that run in fixed-parameter time. Moreover, we show that we cannot do better in FPT time, modulo recent complexity-theoretic conjectures.