In this article, we study approximation algorithms for the problem of computing minimum dominating set for a given set [Formula: see text] of [Formula: see text] unit disks in [Formula: see text]. We first present a simple [Formula: see text] time 5-factor approximation algorithm for this problem, where [Formula: see text] is the size of the output. The best known 4-factor and 3-factor approximation algorithms for the same problem run in time [Formula: see text] and [Formula: see text] 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 [Formula: see text]. A minor modification of this algorithm produces a [Formula: see text]-factor approximation algorithm in [Formula: see text] time. The same techniques can be applied to have a 3-factor and a [Formula: see text]-factor approximation algorithms in time [Formula: see text] and [Formula: see text] respectively. Finally, we propose a very important shifting lemma, which is of independent interest, and it helps to present [Formula: see text]-factor approximation algorithm for the same problem. It also helps to improve the time complexity of the proposed PTAS for the problem.

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In this article, we consider the problem of computing minimum dominating set for a given set S of n points in R 2 . Here the objective is to find a minimum cardinality subset S ′ of S such that the union of the unit radius disks centered at the points in S ′ covers all the points in S. We first propose a simple 4-factor and 3-factor approximation algorithms in O(n 6 log n) and O(n 11 log n) time respectively improving time complexities by a factor of O(n 2 ) and O(n 4 ) respectively over the best known result available in the literature [M. De, G.K. Das, P. Carmi and S.C. Nandy, Approximation algorithms for a variant of discrete piercing set problem for unit disk, Int. J. of Comp. Geom. and Appl., to appear]. Finally, we propose a very important shifting lemma, which is of independent interest and using this lemma we propose a 5 2 -factor approximation algorithm and a PTAS for the minimum dominating set problem.