The burning number of a graph $G$ is the smallest number $b$ such that the vertices of $G$ can be covered by balls of radii $0, 1, \dots, b-1$. As computing the burning number of a graph is known to be NP-hard, even on trees, it is natural to consider polynomial time approximation algorithms for the quantity. The best known approximation factor in the literature is $3$ for general graphs and $2$ for trees. In this note we give a $2/(1-e^{-2})+\varepsilon=2.313\dots$-approximation algorithm for the burning number of general graphs, and a PTAS for the burning number of trees and forests. Moreover, we show that computing a $(\frac43-\varepsilon)$-approximation of the burning number of a general graph $G$ is NP-hard.

翻译：图$G$的燃烧数是满足以下条件的最小整数$b$：$G$的顶点可以被半径分别为$0, 1, \dots, b-1$的球覆盖。已知计算图的燃烧数是NP难的，即使在树上也是如此，因此很自然地考虑该量的多项式时间近似算法。文献中已知的最佳近似因子对于一般图为$3$，对于树为$2$。在本文中，我们给出了一般图燃烧数的$2/(1-e^{-2})+\varepsilon=2.313\dots$近似算法，以及树和森林燃烧数的PTAS。此外，我们证明计算一般图$G$燃烧数的$(\frac43-\varepsilon)$近似是NP难的。