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 $(\lfloor 1.75b(T) \rfloor + 1)/b(T)$.
翻译:根据 $G = (V, E) 图形, \ gb ⁇ 的问题在于从 $V 中找到一个节点序列, 称为燃烧序列, 以烧毁整张图。 这是一个离散的一步过程, 每一步中, 一个未燃烧的顶点被选为一种代理, 将火焰传播到邻居, 标记为烧焦的节点。 一个烧焦的节点将火焰传播到邻居, 目标是找到最小长度的燃烧序列 。 问题在于一般图表甚至二进制树, 问题在于 NP- Hard 。 有几个近似结果, 包括一般图的30美元接近算法和树木的$2美元近似算法。 在本文中, 我们提议了一个树的近似算法, 其燃烧序列以1. 75b (T)\ r 底线 + 1 美元为底部, 其中的长度为$b (T), 也称为 $T 美元 。