A Stackelberg Vertex Cover game is played on an undirected graph $\mathcal{G}$ where some of the vertices are under the control of a \emph{leader}. The remaining vertices are assigned a fixed weight. The game is played in two stages. First, the leader chooses prices for the vertices under her control. Afterward, the second player, called \emph{follower}, selects a min weight vertex cover in the resulting weighted graph. That is, the follower selects a subset of vertices $C^*$ such that every edge has at least one endpoint in $C^*$ of minimum weight w.r.t.\ to the fixed weights, and the prices set by the leader. Stackelberg Vertex Cover (StackVC) describes the leader's optimization problem to select prices in the first stage of the game so as to maximize her revenue, which is the cumulative price of all her (priceable) vertices that are contained in the follower's solution. Previous research showed that StackVC is \textsf{NP}-hard on bipartite graphs, but solvable in polynomial time in the special case of bipartite graphs, where all priceable vertices belong to the same side of the bipartition. In this paper, we investigate StackVC on paths and present a dynamic program with linear time and space complexity.

翻译：Stackelberg顶点覆盖博弈在无向图$\mathcal{G}$上进行，其中部分顶点由\textemph{领导者}控制，其余顶点具有固定权重。该博弈分两个阶段进行：首先，领导者为其控制下的顶点设定价格；随后，称为\textemph{追随者}的第二参与者在生成的赋权图中选择最小权顶点覆盖。即，追随者选取顶点子集$C^*$，使得每条边至少有一个端点在$C^*$中，且该子集相对于固定权重及领导者设定的价格具有最小权值。Stackelberg顶点覆盖（StackVC）描述了领导者在博弈第一阶段的价格选择优化问题，旨在最大化其收益——即追随者解中包含的所有其（可定价）顶点的累计价格。先前研究表明，StackVC在二部图上为\textsf{NP}-难问题，但在所有可定价顶点均属于二部划分同一侧的特殊二部图中可在多项式时间内求解。本文研究路径上的StackVC问题，并提出具有线性时空复杂度的动态规划算法。