Parameterized k-Clustering: The distance matters!

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].