Advancing firefighter games: novel integer programming formulations and the cost-value model

Abstract In the classical firefighter game, a fire breaks out on some vertices of an undirected connected graph at time zero. At each subsequent time step, a fixed number of firefighters can protect one vertex each from catching fire. Afterwards, the fire spreads from each burning vertex to every adjacent vertex that is neither burning nor defended. The game ends when the fire can no longer spread. The goal is to find a defense strategy that maximizes the number of non-burning (saved) vertices. In this work, we first revisit the classical integer linear programming formulation and then present several improvements for it, as well as two new formulations and tighter bounds on the maximum duration of the game. Moreover, we relax the classical assumptions that all vertices have uniform values and costs, i.e., we allow vertices to have different values and costs for being defended. Furthermore, instead of a fixed number of firefighters, we are given a defense budget that we can spend each time step to defend the vertices. We call this the cost-value firefighter game. We present three different integer linear programming formulations for the problem, along with a series of inequalities to strengthen the formulations and tight bounds on the maximum duration of the game.