Improved and Generalized Algorithms for Burning a Planar Point Set

Prashant Gokhale, J. Mark Keil, Debajyoti Mondal

Abstract:

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 + ε) and (1.92188 + ε) approximable, respectively, for every ε > 0, which
improves the previously known (2 + ε) 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.