Geometric Firefighting in the Half-plane

TL;DR

This paper investigates optimal strategies for containing a spreading fire in the half-plane using barriers, demonstrating that delaying barriers can be beneficial and establishing bounds on the necessary fire containment speed.

Contribution

It introduces a new model with delaying barriers in the half-plane and provides improved bounds on the minimum speed required for fire containment.

Findings

Speed v=1.8772 is sufficient for containment.

Speed v>1.66 is necessary for containment.

Delaying barriers can be effective when properly placed.

Abstract

In 2006, Alberto Bressan 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. {\em 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.