On the Approximation of Unit Disk Graph Coordinates

In this paper we study a problem occuring in the context of geometric routing algorithms for mobile ad-hoc networks: Finding unit disk graph coordinates given a graph G = (V, E). Based on a proof that recognition of unit disk graphs is an NP-hard problem, we show that the problem of finding unit disk graph coordinates given a graph G = (V, E) is NPhard. Subsequently, we show that the proof does not extend to quasi unit disk graphs, a generalization of unit disk graphs. We give an exact formulation of the problem of finding unit disk graph coordinates in terms of a quadratic feasibility problem and explore different approximations in terms of linear programs, quadratic programs and semidefinite programs.