Capacitated Vertex Cover is the hard-capacitated variant of Vertex Cover: given a graph, a capacity for every vertex, and an integer $k$, the task is to select at most $k$ vertices that cover all edges and assign each edge to one of its chosen endpoints so that no chosen vertex receives more incident edges than its capacity. This problem is a classical benchmark in parameterized complexity, as it was among the first natural problems shown to be W[1]-hard when parameterized by treewidth. We revisit its exact complexity from a fine-grained parameterized perspective and obtain a much sharper picture for several standard parameters. For the natural parameter $k$, we prove under the Exponential Time Hypothesis (ETH) that no algorithm with running time $k^{o(k)} n^{\mathcal{O}(1)}$ exists. In particular, this shows that the known algorithms with running time $k^{\mathcal{O}(\mathrm{tw})} n^{\mathcal{O}(1)}$ are essentially optimal. We then turn to more general structural parameters. For vertex cover number $\mathrm{vc}$, we give evidence against a $2^{\mathcal{O}(\mathrm{vc}^{2-\varepsilon})} n^{\mathcal{O}(1)}$ algorithm, as such an improvement would imply corresponding progress for a broader class of integer-programming-type problems. We complement this barrier with a nearly matching upper bound for vertex integrity $\mathrm{vi}$, improving the previously known double-exponential dependence to an algorithm with running time $\mathrm{vi}^{\mathcal{O}(\mathrm{vi}^{2})} n^{\mathcal{O}(1)}$ using $N$-fold integer programming. For treewidth, we show that the standard dynamic programming algorithm with running time $n^{\mathcal{O}(\mathrm{tw})}$ is essentially optimal under the ETH, even if one parameterizes by tree-depth. Turning to clique-width, we prove that Capacitated Vertex Cover remains NP-hard already on graphs of linear clique-width $6$...

翻译：容量顶点覆盖是顶点覆盖问题中带硬容量约束的变种：给定一个图、每个顶点的容量以及整数$k$，任务是在最多选择$k$个顶点覆盖所有边的前提下，将每条边分配至其一个被选端点，使得每个被选顶点接收的关联边数不超过其容量。该问题是参数化复杂度领域的经典基准问题，因为它是最早被证明当以树宽为参数时属于W[1]-困难的自然问题之一。我们从细粒度参数化视角重新审视其确切复杂度，针对若干标准参数获得了更清晰的图景。对于自然参数$k$，我们在指数时间假设（ETH）下证明不存在时间复杂度为$k^{o(k)} n^{\mathcal{O}(1)}$的算法，从而表明已知时间复杂度为$k^{\mathcal{O}(\mathrm{tw})} n^{\mathcal{O}(1)}$的算法本质上已达最优。随后转向更一般的结构参数：针对顶点覆盖数$\mathrm{vc}$，我们提供了反对$2^{\mathcal{O}(\mathrm{vc}^{2-\varepsilon})} n^{\mathcal{O}(1)}$算法的证据，因其改进将推动更广泛整数规划类问题的相应进展。与此下界互补，我们针对顶点完整性参数$\mathrm{vi}$给出了几乎匹配的上界，通过$N$重整数规划将先前已知的双指数依赖关系改进为时间复杂度$\mathrm{vi}^{\mathcal{O}(\mathrm{vi}^{2})} n^{\mathcal{O}(1)}$的算法。对于树宽，我们证明即使在参数化为树深度的情况下，标准动态规划算法$n^{\mathcal{O}(\mathrm{tw})}$在ETH下本质上最优。转向团宽时，我们证明容量顶点覆盖在团宽为6的线性团宽图上已属于NP-困难...