Parameterized kClustering: The distance matters!
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 \[\sum_{i=1}^k \min_{c_i\in \mathbb{R}^d}\sum_{x \in C_i} \xc_i\_p^p \leq D,\] where $\\cdot\_p$ is the Minkowski ($L_p$) norm of order $p$. For $p=1$, $k$Clustering is the wellknown $k$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 fixedparameter 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].
 Publication:

arXiv eprints
 Pub Date:
 February 2019
 arXiv:
 arXiv:1902.08559
 Bibcode:
 2019arXiv190208559F
 Keywords:

 Computer Science  Data Structures and Algorithms