# Exact FPT Algorithm for Continuous Euclidean $k$-Means

The continuous Euclidean $$k$$-means problem is defined as follows:

Given a set $$X$$ of $$n$$ points in $$d$$ dimensional Euclidean space $$\mathbb{R}^{d}$$. Given a parameter $$k>0$$, find a partitioning $$P$$ of $$X$$ into sets $$X_1,\dotsc,X_k$$ such that the following clustering cost is minimized:

$$\Phi(P) = \sum_{i = 1}^{k} \sum_{x \in X_i} \|x - \mu(X_i)\|^{2}, \quad \textrm{where \mu(X_{i}) is the centroid of X_i.}$$

For constant $$k$$, the problem is $$\mathsf{NP}$$-hard (see here).

For constant $$d$$, the problem is $$\mathsf{NP}$$-hard (see here).

What about when both $$k$$ and $$d$$ are constant? Well, there is an $$O(n^{kd})$$ algorithm for the problem (see here). But is there any exact $$\mathsf{FPT}$$ algorithm known for the problem with running time $$f(k,d) \cdot n^{O(1)}$$ or any related $$\mathsf{FPT}$$ hardness result known?

Note: I know that discrete metric $$k$$-means problem is $$\mathsf{W}[2]$$-hard, parametrized by $$k$$ (see here). But it does not imply $$\mathsf{FPT}$$ hardness for the continuous Euclidean space.