Minimum Dominating Set Problem for Unit Disks Revisited

In this article, we study approximation algorithms for the problem of computing minimum dominating set for a given set S of n unit disks in ℝ2. We first present a simple O(n log k) time 5-factor approximation algorithm for this problem, where k is the size of the output. The best known 4-factor and 3-factor approximation algorithms for the same problem run in time O(n8 log n) and O(n15 log n) respectively [M. De, G. K. Das, P. Carmi and S. C. Nandy, Approximation algorithms for a variant of discrete piercing set problem for unit disks, Int. J. of Computational Geometry and Appl., 22(6):461–477, 2013]. We show that the time complexity of the in-place 4-factor approximation algorithm for this problem can be improved to O(n6 log n). A minor modification of this algorithm produces a 143-factor approximation algorithm in O(n5 log n) time. The same techniques can be applied to have a 3-factor and a 4513-factor approximation algorithms in time O(n11 log n) and O(n10 log n) respectively. Finally, we propose a very important shifting lemma, which is of independent interest, and it helps to present 52-factor approximation algorithm for the same problem. It also helps to improve the time complexity of the proposed PTAS for the problem.