  We develop an FPT algorithm and a bi-kernel for the Weighted Edge Clique
Partition (WECP) problem, where a graph with $n$ vertices and integer edge
weights is given together with an integer $k$, and the aim is to find $k$
cliques, such that every edge appears in exactly as many cliques as its weight.
The problem has been previously only studied in the unweighted version called
Edge Clique Partition (ECP), where the edges need to be partitioned into $k$
cliques. It was shown that ECP admits a kernel with~$k^2$ vertices [Mujuni and
Rosamond, 2008], but this kernel does not extend to WECP. The previously
fastest algorithm known for ECP has a runtime of $2^{\mathcal{O}(k^2)}n^{O(1)}$
[Issac, 2019]. For WECP we develop a bi-kernel with $4^k$ vertices, and an
algorithm with runtime $2^{\mathcal{O}(k^{3/2}w^{1/2}\log(k/w))}n^{O(1)}$,
where $w$ is the maximum edge weight. The latter in particular improves the
runtime for ECP to~$2^{\mathcal{O}(k^{3/2}\log k)}n^{O(1)}$.
