Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions

TL;DR

This paper studies how to partition graphs and polygons into greedily routable regions, providing complexity results and optimal algorithms for trees, and approximation methods for polygons.

Contribution

It proves NP-hardness for cycle graphs, offers polynomial-time solutions for trees, and presents a 2-approximation for polygons with fixed triangulations.

Findings

NP-hardness for graphs with cycles

Polynomial-time algorithm for trees

2-approximation for polygons with given triangulation

Abstract

A greedily routable region (GRR) is a closed subset of $\mathbb R^2$, in which each destination point can be reached from each starting point by choosing the direction with maximum reduction of the distance to the destination in each point of the path. Recently, Tan and Kermarrec proposed a geographic routing protocol for dense wireless sensor networks based on decomposing the network area into a small number of interior-disjoint GRRs. They showed that minimum decomposition is NP-hard for polygons with holes. We consider minimum GRR decomposition for plane straight-line drawings of graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing style which has become a popular research topic in graph drawing. We show that minimum decomposition is still NP-hard for graphs with cycles, but can be solved optimally for trees in polynomial time. Additionally, we give a 2-approximation for simple polygons, if a given triangulation has to be respected.