Improved Approximation Algorithm for Graph Burning on Trees

Given a graph $G=(V,E)$, the problem of \gb{} is to find a sequence of nodes from $V$, called burning sequence, in order to burn the whole graph. This is a discrete-step process, in each step an unburned vertex is selected as an agent to spread fire to its neighbors by marking it as a burnt node. A node that is burnt spreads the fire to its neighbors at the next consecutive step. The goal is to find the burning sequence of minimum length. The \gb{} problem is NP-Hard for general graphs and even for binary trees. A few approximation results are known, including a $3$-approximation algorithm for general graphs and a $2$- approximation algorithm for trees. In this paper, we propose an approximation algorithm for trees that produces a burning sequence of length at most $\lfloor 1.75b(T) \rfloor + 1$, where $b(T)$ is length of the optimal burning sequence, also called the burning number of the tree $T$. In other words, we achieve an approximation factor of $(\lfloor 1.75b(T) \rfloor + 1)/b(T)$.