APX-Hardness and Approximation for the k-Burning Number Problem

Consider an information diffusion process on a graph G that starts with \(k>0\) burnt vertices, and at each subsequent step, burns the neighbors of the currently burnt vertices, as well as k other unburnt vertices. The k-burning number of G is the minimum number of steps \(b_k(G)\) such that all the vertices can be burned within \(b_k(G)\) steps. Note that the last step may have smaller than k unburnt vertices available, where all of them are burned. The 1-burning number coincides with the well-known burning number problem, which was proposed to model the spread of social contagion. The generalization to k-burning number allows us to examine different worst-case contagion scenarios by varying the spread factor k.