Given an undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, with vertex weights $(w(u))_{u\in\mathcal{V}}$, vertex values $(\alpha(u))_{u\in\mathcal{V}}$, a knapsack size $s$, and a target value $d$, the \vcknapsack problem is to determine if there exists a subset $\mathcal{U}\subseteq\mathcal{V}$ of vertices such that $\mathcal{U}$ forms a vertex cover, $w(\mathcal{U})=\sum_{u\in\mathcal{U}} w(u) \le s$, and $\alpha(\mathcal{U})=\sum_{u\in\mathcal{U}} \alpha(u) \ge d$. In this paper, we closely study the \vcknapsack problem and its variations, such as \vcknapsackbudget, \minimalvcknapsack, and \minimumvcknapsack, for both general graphs and trees. We first prove that the \vcknapsack problem belongs to the complexity class \NPC and then study the complexity of the other variations. We generalize the problem to \setc and \hs versions and design polynomial time $H_g$-factor approximation algorithm for the \setckp problem and d-factor approximation algorithm for \hstp using primal dual method. We further show that \setcks and \hsmb are hard to approximate in polynomial time. Additionally, we develop a fixed parameter tractable algorithm running in time $8^{\mathcal{O}({\rm tw})}\cdot n\cdot {\sf min}\{s,d\}$ where ${\rm tw},s,d,n$ are respectively treewidth of the graph, the size of the knapsack, the target value of the knapsack, and the number of items for the \minimalvcknapsack problem.

翻译：给定一个无向图 $\mathcal{G}=(\mathcal{V},\mathcal{E})$，其顶点权重为 $(w(u))_{u\in\mathcal{V}}$，顶点价值为 $(\alpha(u))_{u\in\mathcal{V}}$，背包容量为 $s$，目标价值为 $d$，\vcknapsack 问题旨在判断是否存在一个顶点子集 $\mathcal{U}\subseteq\mathcal{V}$，使得 $\mathcal{U}$ 构成一个顶点覆盖，$w(\mathcal{U})=\sum_{u\in\mathcal{U}} w(u) \le s$，且 $\alpha(\mathcal{U})=\sum_{u\in\mathcal{U}} \alpha(u) \ge d$。本文深入研究了 \vcknapsack 问题及其变体，例如 \vcknapsackbudget、\minimalvcknapsack 和 \minimumvcknapsack，涵盖一般图和树结构。我们首先证明了 \vcknapsack 问题属于复杂度类 \NPC，随后研究了其他变体的复杂度。我们将该问题推广至 \setc 和 \hs 版本，并利用原始对偶方法为 \setckp 问题设计了多项式时间的 $H_g$ 因子近似算法，为 \hstp 问题设计了 d 因子近似算法。我们进一步证明了 \setcks 和 \hsmb 问题在多项式时间内难以近似。此外，我们针对 \minimalvcknapsack 问题开发了一种固定参数可处理算法，其运行时间为 $8^{\mathcal{O}({\rm tw})}\cdot n\cdot {\sf min}\{s,d\}$，其中 ${\rm tw}, s, d, n$ 分别为图的树宽、背包容量、背包目标价值和物品数量。