Geometric Firefighting in the Half-Plane

Abstract

In 2006, Alberto Bressan [3] suggested the following problem. Suppose a circular fire spreads in the Euclidean plane at unit speed. The task is to build, in real time, barrier curves to contain the fire. At each time t the total length of all barriers built so far must not exceed $t \cdot v$, where v is a speed constant. How large a speed v is needed? He proved that speed $v>2$ is sufficient, and that $v>1$ is necessary. This gap of (1, 2] is still open. The crucial question seems to be the following. When trying to contain a fire, should one build, at maximum speed, the enclosing barrier, or does it make sense to spend some time on placing extra delaying barriers in the fire’s way? We study the situation where the fire must be contained in the upper $L_1$ half-plane by an infinite horizontal barrier to which vertical line segments may be attached as delaying barriers. Surprisingly, such delaying barriers are helpful when properly placed. We prove that speed $v=1.8772$ is sufficient, while $v >1.66$ is necessary.

This work has been supported by DFG grant Kl 655/19 as part of a DACH project.