Minimum Dominating Set for a Point Set in $\IR^2$

In this article, we consider the problem of computing minimum dominating set for a given set $S$ of $n$ points in $\IR^2$. 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, {\it 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 $\frac{5}{2}$-factor approximation algorithm and a PTAS for the minimum dominating set problem.