A vertex set $W$ in a graph $G$ is a valid $k$-swap for a vertex cover $S$ of $G$ if $W$ has size at most $k$ and $S'=(S \setminus W) \cup (W \setminus S)$, the symmetric difference of $S$ and $W$, is a vertex cover of $G$. If $|S'| < |S|$, then $W$ is improving. In LS Vertex Cover, one is given a vertex cover $S$ of a graph $G$ and wants to know if there is a valid improving $k$-swap for $S$ in $G$. In applications of LS Vertex Cover, $k$ is a very small parameter that can be set by a user to determine the trade-off between running time and solution quality. Consequently, $k$ can be considered to be a constant. Motivated by this and the fact that LS Vertex Cover is W[1]-hard with respect to $k$, we aim for algorithms with running time $\ell^{f(k)}\cdot n^{\mathcal{O}(1)}$ where $\ell$ is a structural graph parameter upper-bounded by $n$. We say that such a running time grows mildly with respect to $\ell$ and strongly with respect to $k$. We obtain algorithms with such a running time for $\ell$ being the $h$-index of $G$, the treewidth of $G$, or the modular-width of $G$. In addition, we consider a novel parameter, the maximum degree over all quotient graphs in a modular decomposition of $G$. Moreover, we adapt these algorithms to the more general problem where each vertex is assigned a weight and where we want to find a valid $d$-improving $k$-swap, that is, a valid $k$-swap which decreases the weight of the vertex cover by at least $d$.

翻译：图$G$中的顶点集$W$是顶点覆盖$S$的有效$k$-交换，若$|W|\leq k$且$S'=(S \setminus W) \cup (W \setminus S)$（即$S$与$W$的对称差）是$G$的一个顶点覆盖。若$|S'|<|S|$，则称$W$为改进性交换。在局部搜索顶点覆盖问题中，给定图$G$的一个顶点覆盖$S$，需判断$G$中是否存在关于$S$的有效改进性$k$-交换。在局部搜索顶点覆盖的应用中，$k$是由用户设定的极小参数，用于权衡运行时间与解质量，因此$k$可视为常数。基于此及局部搜索顶点覆盖关于$k$为W[1]-难的特性，我们致力于设计运行时间为$\ell^{f(k)}\cdot n^{\mathcal{O}(1)}$的算法，其中$\ell$是受$n$限制的结构图参数。此类运行时间对$\ell$具有温和增长性，而对$k$呈强增长性。当$\ell$分别为图$G$的$h$-指数、树宽或模宽度时，我们获得了具有上述运行时间的算法。此外，我们引入一个新参数：$G$的模分解中所有商图的最大度。进一步，我们将这些算法推广至更一般的问题：每个顶点被赋予权重，需寻找有效$d$-改进性$k$-交换，即能将顶点覆盖权重至少降低$d$的有效$k$-交换。