Packing and Covering a Polygon with Geodesic Disks

Abstract

Given a polygon $P$, for two points $s$ and $t$ contained in the polygon, their \emph{geodesic distance} is the length of the shortest $st$-path within $P$. A \emph{geodesic disk} of radius $r$ centered at a point $v\in P$ is the set of points in $P$ whose geodesic distance to $v$ is at most $r$. We present a polynomial time $2$-approximation algorithm for finding a densest geodesic unit disk packing in $P$. Allowing arbitrary radii but constraining the number of disks to be $k$, we present a $4$-approximation algorithm for finding a packing in $P$ with $k$ geodesic disks whose minimum radius is maximized. We then turn our focus on \emph{coverings} of $P$ and present a $2$-approximation algorithm for covering $P$ with $k$ geodesic disks whose maximal radius is minimized. Furthermore, we show that all these problems are $\mathsf{NP}$-hard in polygons with holes. Lastly, we present a polynomial time exact algorithm which covers a polygon with two geodesic disks of minimum maximal radius.

References in corpus (2)

• Point Set Isolation Using Unit Disks is NP-complete
• Packing identical simple polygons is NP-hard

Cited by in corpus (3)

• Packing and covering with balls on Busemann surfaces
• Covering the Boundary of a Simple Polygon with Geodesic Unit Disks
• Approximation algorithms for the two-center problem of convex polygon