Fixed Parameter Tractability of Weighted Edge Clique Partition
Conference: International Symposium on Parameterized and Exact Computation 2020
Speaker: Davis Issac
Abstract: We develop an FPT algorithm and a compression 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 \(2^{O(k^2)}n^{O(1)}\) [Issac, 2019 (PhD Thesis)]. For WECP we develop a compression (to a slightly more general problem called Annotated WECP) with at most \(4^k\) 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 \(2^{O(k^{3/2}\log k)}n^{O(1)}\).
This is joined work with Andreas Emil Feldmann and Ashutosh Rai.