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顶点覆盖博弈在一个无向图$\mathcal{G}$上进行，其中部分顶点由“领导者”控制，其余顶点被赋予固定权重。该博弈分为两个阶段进行：首先，领导者为其控制的顶点设定价格；随后，第二玩家（称为“跟随者”）在生成的加权图中选择一个最小权重顶点覆盖。即，跟随者选择一个顶点子集$C^*$，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和领导者设定的价格具有最小权重。Stackelberg顶点覆盖（StackVC）描述了领导者在博弈第一阶段通过选择价格以最大化自身收益的优化问题——其收益为跟随者解中所有被定价顶点（即控制权属领导者的顶点）的累计价格。此前研究表明，StackVC在二分图上是$\textsf{NP}$-难的，但在所有可定价顶点均属于二分图同一侧的二分图特例中，可在多项式时间内求解。本文研究路径上的StackVC问题，并提出一种具有线性时间与空间复杂度的动态规划算法。