# Parameterized k-Clustering: The distance matters!

## TL;DR

This paper investigates the parameterized complexity of the k-Clustering problem under different Minkowski distances, revealing tractability for p in (0,1] and hardness for p=0 and p=∞.

## Contribution

It establishes the fixed-parameter tractability of k-Clustering for p in (0,1], and proves hardness results for p=0 and p=∞, highlighting the importance of distance choice.

## Key findings

- FPT algorithm for p in (0,1] with runtime 2^{O(D log D)}(nd)^{O(1)}.
- Hardness results for p=0 and p=∞, unless FPT=W[1].
- Distance order p critically affects the complexity of k-Clustering.

## Abstract

We consider the $k$-Clustering problem, which is for a given multiset of $n$ vectors $X\subset \mathbb{Z}^d$ and a nonnegative number $D$, to decide whether $X$ can be partitioned into $k$ clusters $C_1, \dots, C_k$ such that the cost \[\sum_{i=1}^k \min_{c_i\in \mathbb{R}^d}\sum_{x \in C_i} \|x-c_i\|_p^p \leq D,\] where $\|\cdot\|_p$ is the Minkowski ($L_p$) norm of order $p$. For $p=1$, $k$-Clustering is the well-known $k$-Median. For $p=2$, the case of the Euclidean distance, $k$-Clustering is $k$-Means. We show that the parameterized complexity of $k$-Clustering strongly depends on the distance order $p$. In particular, we prove that for every $p\in (0,1]$, $k$-Clustering is solvable in time $2^{O(D \log{D})} (nd)^{O(1)}$, and hence is fixed-parameter tractable when parameterized by $D$. On the other hand, we prove that for distances of orders $p=0$ and $p=\infty$, no such algorithm exists, unless FPT=W[1].