Burning Number for the Points in the Plane

TL;DR

This paper studies the computational complexity of a geometric variant of the graph burning problem, proving NP-completeness and approximation bounds for point burning and anywhere burning in the Euclidean plane.

Contribution

It introduces point burning and anywhere burning variants, establishes their NP-completeness, and provides approximation algorithms and hardness results.

Findings

Both point burning and anywhere burning are NP-complete.

Both problems are $(2+ ext{epsilon})$-approximable for any epsilon > 0.

Approximation hardness for restricted anywhere burning is at least $rac{2}{\sqrt{3}}- ext{epsilon}$.

Abstract

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 $\frac{2}{\sqrt{3}}-\varepsilon$.