Given a set $P$ of points in the plane, a point burning process is a discrete time process to burn all the points of $P$ where fires must be initiated at the given points. Specifically, the point burning process starts with a single burnt point from $P$, and at each subsequent step, burns all the points in the plane that are within one unit distance from the currently burnt points, as well as one other unburnt point of $P$ (if exists). The point burning number of $P$ is the smallest number of steps required to burn all the points of $P$. If we allow the fire to be initiated anywhere, then the burning process is called an anywhere burning process, and the corresponding burning number is called anywhere burning number. Computing the point and anywhere burning number is known to be NP-hard. In this paper we show that both these problems admit PTAS in one dimension. We then show that in two dimensions, point burning and anywhere burning are $(1.96296+\varepsilon)$ and $(1.92188+\varepsilon)$ approximable, respectively, for every $\varepsilon>0$, which improves the previously known $(2+\varepsilon)$ factor for these problems. We also observe that a known result on set cover problem can be leveraged to obtain a 2-approximation for burning the maximum number of points in a given number of steps. We show how the results generalize if we allow the points to have different fire spreading rates. Finally, we prove that even if the burning sources are given as input, finding a point burning sequence itself is NP-hard.
翻译:鉴于平面点数的设定值为$P美元,点燃烧过程是一个离散的时间过程,可以燃烧所有点点点,因为必须在给定点点点点点起火。具体地说,点燃烧过程从一个点点点起,从一个点点起,然后在每个步骤点点起,烧掉飞机上与目前燃烧点距离内的所有点,以及另一个未燃烧点点点点点(如果存在的话),点燃烧次数是燃烧所有点点点点点(P$)所需的最小步骤数。如果我们允许在任何地点点点起火,那么燃烧过程就叫一个不同点点点点点点点起,而相应的燃烧次数则在任何燃烧点点点点点点点点点点点点起,已知点点点点点点点点点点点点点点点点点点点是NPP-硬的。在本文中显示,在两个层面,点点点点点点点点点点点点点点烧和点点烧点数是给定的(P),如果我们给出了1.988-瓦列普斯朗)一般点点点点点点点点点点点点点起火,,那么我们的燃烧速度也可以控制点点点点点点点点点点点点点点点点点点点点点点点点点点点点点点点点点点点点点点点点点数。