Fixed-Parameter Tractability of the Weighted Edge Clique Partition Problem.

We develop an FPT algorithm and a 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 had a runtime of $2^{O(k^2)}n^{O(1)}$ [Issac, 2019]. For WECP we develop a bi-kernel with $4^k$ vertices, and an algorithm with runtime $2^{O(k^{3/2}w^{1/2}\log(k/w))}n^{O(1)}$, where $w$ is the maximum edge weight. The latter in particular improves the runtime for ECP to~$2^{O(k^{3/2}\log k)}n^{O(1)}$. We also show that our algorithm necessarily needs a runtime of $2^{\Theta(k^{3/2}\log k)}n^{O(1)}$ to solve ECP.