Stackelberg Vertex Cover on a Path

TL;DR

This paper studies the Stackelberg Vertex Cover game on paths, providing a linear-time dynamic programming solution for the leader's optimal pricing problem in this specific graph class.

Contribution

It introduces a novel linear-time dynamic programming algorithm for solving StackVC on path graphs, expanding understanding of this game beyond bipartite cases.

Findings

Linear-time algorithm for StackVC on paths

Efficient computation of optimal leader prices

Extension of StackVC analysis to path graphs

Abstract

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.