In a large-scale network, we want to choose some influential nodes to make a profit by paying some cost within a limited budget so that we do not have to spend more budget on some nodes adjacent to the chosen nodes; our problem is the graph-theoretic representation of it. We define our problem, Dominating Set Knapsack, by attaching the knapsack problem with the dominating set on graphs. Each vertex $v~(\in V) $ is associated with a cost factor $w(v)$ and a profit amount $α(v)$. We aim to choose some vertices within a fixed budget $(s)$ that give maximum profit so that we do not need to choose their 1-hop neighbors. We show that the Dominating Set Knapsack problem is strongly NPC even when restricted to bipartite graphs, but weakly NPC for star graphs. We present a pseudo-polynomial time algorithm for trees in time $O(n\cdot min\{s^2, (α(V))^2\})$. We show that Dominating Set Knapsack is unlikely to be Fixed Parameter Tractable (FPT) by proving that it is W[2]-hard parameterized by the solution size. We developed FPT algorithms with running time $O(4^{tw}\cdot n^{O(1)} min\{s^2,{α(V)}^2\})$ and $O(2^{vck-1}\cdot n^{O(1)} min\{s^2,{α(V)}^2\})$, where $tw$ represents the $tw$ of the given graph $G(V,E)$, $vck$ is the solution size of the Vertex Cover Knapsack, $s$ is the capacity or size of the knapsack and $α(V)=\sum_{v\in V}α(v)$. We obtained similar results for other variants $k-$Dominating Set Knapsack and Minimal Dominating Set Knapsack, where $k$ is the size of the dominating set.

翻译：在大规模网络中，我们希望通过在有限预算内支付一定成本来选择部分有影响力的节点以获取利润，从而避免为与所选节点相邻的某些节点额外支出预算；我们的问题即是该场景的图论表示。我们通过将背包问题与图上的支配集相结合，定义了支配集背包问题。每个顶点$v~(\in V)$关联一个成本因子$w(v)$和一个利润值$α(v)$。我们的目标是在固定预算$(s)$内选择部分顶点以获得最大利润，同时确保无需选择其一跳邻居。我们证明支配集背包问题即使在二分图上也是强NPC问题，但在星形图上为弱NPC问题。我们提出了一种针对树结构的伪多项式时间算法，时间复杂度为$O(n\cdot min\{s^2, (α(V))^2\})$。通过证明该问题在解大小参数化下具有W[2]难度，我们表明支配集背包问题不太可能是固定参数可解的。我们开发了运行时间为$O(4^{tw}\cdot n^{O(1)} min\{s^2,{α(V)}^2\})$和$O(2^{vck-1}\cdot n^{O(1)} min\{s^2,{α(V)}^2\})$的FPT算法，其中$tw$表示给定图$G(V,E)$的树宽，$vck$是顶点覆盖背包问题的解大小，$s$是背包容量或大小，$α(V)=\sum_{v\in V}α(v)$。对于其他变体$k-$支配集背包问题和极小支配集背包问题（其中$k$为支配集大小），我们也获得了类似的结果。