A fixed-parameter tractable algorithm for matrix domination
Information Processing Letters
Invitation to data reduction and problem kernelization
ACM SIGACT News
New parameterized algorithms for the edge dominating set problem
MFCS'11 Proceedings of the 36th international conference on Mathematical foundations of computer science
EDGE DOMINATING SET: efficient enumeration-based exact algorithms
IWPEC'06 Proceedings of the Second international conference on Parameterized and Exact Computation
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A kernelization for a parameterized computational problem is a polynomial-time procedure that transforms every instance of the problem into an equivalent instance (the so-called kernel) whose size is bounded by a function of the value of the chosen parameter. We present new kernelizations for the NP-complete Edge Dominating Set problem which asks, given an undirected graph G=(V,E) and an integer k, whether there exists a subset D⊆E with |D|≤k such that every edge in E shares at least one endpoint with some edge in D. The best previous kernelization for Edge Dominating Set, due to Xiao, Kloks and Poon, yields a kernel with at most 2 k2+2 k vertices in linear time. We first describe a very simple linear-time kernelization whose output has at most 4 k2+4 k vertices and is either a trivial "no" instance or a vertex-induced subgraph of the input graph in which every edge dominating set of size ≤k is also an edge dominating set of the input graph. We then show that a refinement of the algorithm of Xiao, Kloks and Poon and a different analysis can lower the bound on the number of vertices in the kernel by a factor of about 4, namely to $\max\{\frac{1}{2}k^2+\frac{7}{2}k,6 k\}$.