Computing the Geodesic Centers of a Polygonal Domain

We present an algorithm that computes the geodesic center of a given polygonal domain. The running time of our algorithm is $O(n^{12+ϵ})$ for any $ϵ>0$, where $n$ is the number of corners of the input polygonal domain. Prior to our work, only the very special case where a simple polygon is given as input has been intensively studied in the 1980s, and an $O(n \log n)$-time algorithm is known by Pollack et al. Our algorithm is the first one that can handle general polygonal domains having one or more polygonal holes.