Approximation Algorithms for Unit Disk Graphs

Mobile ad hoc networks are frequently modeled by unit disk graphs. We consider several classical graph theoretic problems on unit disk graphs (Maximum Independent Set, Minimum Vertex Cover, and Minimum (Connected) Dominating Set), which are relevant to such networks. We propose two new notions for unit disk graphs: thickness and density. The thickness of a graph is the number of disk centers in any width 1 slab. If the thickness of a graph is bounded, then the considered problems can be solved in polynomial time. We prove this both indirectly by presenting a relation between unit disk graphs of bounded thickness and the pathwidth of such graphs, and directly by giving dynamic programming algorithms. This result implies that the problems are fixed-parameter tractable in the thickness. We then consider unit disk graphs of bounded density. The density of a graph is the number of disk centers in any 1-by-1 box. We present a new approximation scheme for the considered problems, which uses the bounded thickness results mentioned above and the so called shifting technique. We show that the scheme is an asymptotic FPTAS and that this result is optimal, in the sense that no FPTAS can exist (unless P=NP). The scheme for Minimum Connected Dominating Set is the first FPTAS∞ for this problem. The analysis that is applied can also be used to improve existing results, which among others implies the existence of an FPTAS∞ for MCDS on planar graphs.