平面上点的燃烧数
Burning Number for the Points in the Plane
摘要：
图$G$上的燃烧过程从单个燃烧顶点开始，并且在每个后续步骤中，燃烧当前燃烧顶点的邻居以及一个其他未燃烧顶点。$G$的燃烧数是燃烧图的所有顶点所需的最小步骤数。在本文中，我们研究的问题，计算燃烧数的几何设置。输入是欧几里德平面中的一组点$P$。燃烧过程从单个燃烧点开始，并且在每个后续步骤中，燃烧距离当前燃烧点和另一个未燃烧点一个单位的距离内的所有点。$P$的燃烧数是燃烧$P$的所有点所需的最小步骤数。我们称这种变异体为\emph{point burning}。我们考虑另一种称为\emph{anywhere burning}的变体，其中允许烧平面的任何点。我们证明了点燃烧和任何地方燃烧问题都是NP完全的，但对每个$\varepsilon>0$都是$(2+\varepsilon)$可逼近的。此外，如果我们对可以使用的刻录源的数量进行限制，那么任何地方刻录问题都将成为NP-难的，难以在$\frac{2}{\sqrt{3}}-\varepsilon$的因子内近似。
英文摘要：
The burning process on a graph $G$ starts with a single burnt vertex, and at each subsequent step, burns the neighbors of the currently burnt vertices, as well as one other unburnt vertex. The burning number of $G$ is the smallest number of steps required to burn all the vertices of the graph. In this paper, we examine the problem of computing the burning number in a geometric setting. The input is a set of points $P$ in the Euclidean plane. The burning process starts with a single burnt point, and at each subsequent step, burns all the points that are within a distance of one unit from the currently burnt points and one other unburnt point. The burning number of $P$ is the smallest number of steps required to burn all the points of $P$. We call this variant \emph{point burning}. We consider another variant called \emph{anywhere burning}, where we are allowed to burn any point of the plane. We show that point burning and anywhere burning problems are both NP-complete, but $(2+\varepsilon)$ approximable for every $\varepsilon>0$. Moreover, if we put a restriction on the number of burning sources that can be used, then the anywhere burning problem becomes NP-hard to approximate within a factor of $\frac{2}{\sqrt{3}}-\varepsilon$.
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