Algorithms for dominating set in disk graphs: Breaking the log n barrier (Extended abstract)

Abstract

We consider the problem of finding a lowest cost dominating set in a given disk graph containing n disks. The problem has been extensively studied on subclasses of disk graphs, yet the best known approximation for disk graphs has remained O(logn) - a bound that is asymptotically no better than the general case. We improve the status quo in two ways: for the unweighted case, we show how to obtain a PTAS using the framework recently proposed (independently) by Mustafa and Ray [16] and by Chan and Har-Peled [4]; for the weighted case where each input disk has an associated rational weight with the objective of finding a minimum cost dominating set, we give a randomized algorithm that obtains a dominating set whose weight is within a factor 2O(log* n) of a minimum cost solution, with high probability - the technique follows the framework proposed recently by Varadarajan [19]. © 2010 Springer-Verlag.