Partitioning polygons via graph augmentation

Abstract

We study graph augmentation under the dilation criterion. In our case, we consider a plane geometric graph G = (V,E) and a set C of edges.We aim to add toGaminimal number of nonintersecting edges from C to bound the ratio between the graph-based distance and the Euclidean distance for all pairs of vertices described by C.Motivated by the problem of decomposing a polygon into natural subregions, we present an optimal linear-time algorithm for the case that P is a simple polygon and C models an internal triangulation of P. The algorithm admits some straightforward extensions. Most importantly, in pseudopolynomial time, it can approximate a solution of minimum total length or, if C is weighted, compute a solution of minimum total weight. We show that minimizing the total length or the total weight is weakly NP-hard. Finally, we show how our algorithm can be used for two well-known problems in GIS: generating variable-scale maps and area aggregation.