Good quality virtual realization of unit disk graphs

Abstract

We consider the problem of finding a realization of an n-vertex unit disk graph (UDG) expressed in general form , say, as an adjacency matrix. The problem is to construct an embedding of the graph in low-dimensional Euclidean space so that the ratio of the length of the longest edge under the embedding to the length of the shortest non-edge under the embedding is as small as possible; the measure is known as the quality of the realization. Thus, an optimum quality realization has quality between 1/2 and 1. Kuhn et al. gave a O (log 3.5 n (loglog n ) 1/2 }) quality realization that requires solving a linear program with exponentially many constraints by using the ellipsoid algorithm. In this article, we give a combinatorial algorithm that achieves an O (log 3 n ) quality realization of an n -vertex UDG expressed in general form. Thus, not only is our algorithm an improvement, it also bypasses the standard and costly technique of solving a linear program with exponentially many “spreading constraints.” As a side effect of our construction, we get the first constant-factor approximation to the minimum clique partition problem for UDGs expressed in general form. Such a clique partition also represents our key technical contribution. If the embedding is allowed to reside in higher dimensional space, we obtain improved results: a quality-2 embedding in constant dimensional Euclidean space.