The Polygon Burning Problem

Motivated by the $k$-center problem in location analysis, we consider the \emph{polygon burning} (PB) problem: Given a polygonal domain $P$ with $h$ holes and $n$ vertices, find a set $S$ of $k$ vertices of $P$ that minimizes the maximum geodesic distance from any point in $P$ to its nearest vertex in $S$. Alternatively, viewing each vertex in $S$ as a site to start a fire, the goal is to select $S$ such that fires burning simultaneously and uniformly from $S$, restricted to $P$, consume $P$ entirely as quickly as possible. We prove that PB is NP-hard when $k$ is arbitrary. We show that the discrete $k$-center of the vertices of $P$ under the geodesic metric on $P$ provides a $2$-approximation for PB, resulting in an $O(n^2 \log n + hkn \log n)$-time $3$-approximation algorithm for PB. Lastly, we define and characterize a new type of polygon, the sliceable polygon. A sliceable polygon is a convex polygon that contains no Voronoi vertex from the Voronoi diagram of its vertices. We give a dynamic programming algorithm to solve PB exactly on a sliceable polygon in $O(kn^2)$ time.