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 $$\mathbb {R}^2$$ , a point burning process is a discrete time process to burn all the points of P where fires must be initiated at the points of P. Specifically, the point burning process starts with a single burnt point from P, and at each subsequent step, burns all the points in $$\mathbb {R}^2$$ 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 in $$\mathbb {R}^2$$ , then the burning process is called an anywhere burning process. One can think of the anywhere burning problem as finding the minimum integer r such that P can be covered with disks of radii $$0,1,2,\ldots , r$$ . A burning process provides a simple model for distributing commodities to the locations in P by sending a daily bulk shipment to a distribution center, i.e., the place where we initiate a fire. The burning number corresponds to the minimum number of days to reach all locations. In this paper we show that both point and anywhere burning problems admit PTAS in one dimension. We then show that in two dimensions, point burning and anywhere burning are $$(1.96+\varepsilon )$$ and $$(1.92+\varepsilon )$$ approximable, respectively, for every $$\varepsilon >0$$ , which improves the previously known $$(2+\varepsilon )$$ approximation for these problems. We then generalize the results by allowing the points to have different fire spreading rates, and prove that even if the burning sources are given as input, finding a point burning sequence itself is NP-hard. Finally, we obtain a 2-approximation for burning the maximum number of points in a given number of steps.