Capacitated Vertex Cover is a generalization of Vertex Cover where we seek a set of $k$ vertices covering all edges, but with the added constraint that for each vertex we are given a capacity, that is, a maximum number of incident edges the vertex can cover if selected. This problem is well-studied in parameterized complexity and is known to be in XP but W[1]-hard parameterized by treewidth. Natural dynamic programs lead to complexities of the form $C^{\mathcal{O}(\mathrm{tw})}n^{\mathcal{O}(1)}$, $Δ^{\mathcal{O}(\mathrm{tw})}n^{\mathcal{O}(1)}$, or $k^{\mathcal{O}(\mathrm{tw})}n^{\mathcal{O}(1)}$, where $C, Δ$ are respectively the maximum capacity and maximum degree. These results imply an FPT algorithm for parameter $k$ with complexity $k^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$, as $\mathrm{tw}\le k$. These simple dynamic programming algorithms essentially constitute the state of the art. Our main question is therefore whether these algorithms can be improved. We address this question from numerous angles...

翻译：暂无翻译