  In this paper, we study two classic optimization problems: minimum geometric
dominating set and set cover. In the dominating-set problem, for a given set of
objects in {the} plane as input, the objective is to choose a minimum number of
input objects such that every input object is dominated by the chosen set of
objects. Here, one object is dominated by {another} if both of them have {a}
nonempty intersection region. For the second problem, for a given set of points
and objects {in a plane}, the objective is to choose {a} minimum number of
objects to cover all the points. This is a special version of the set-cover
problem.
  For both problems obtaining a PTAS remains open for a large class of objects.
  For the dominating-set problem, we prove that {a} popular local-search
algorithm leads to an $(1+\varepsilon)$ approximation for object sets
consisting of homothetic set of convex objects (which includes arbitrary
squares, $k$-regular polygons, translated and scaled copies of a convex set,
etc.) in $n^{O(1/\varepsilon^2)}$ time. On the other hand, the same technique
leads to a PTAS for geometric covering problem when the objects are convex
pseudodisks (which includes disks, unit height rectangles, homothetic convex
objects, etc.). As a consequence, we obtain an easy to implement approximation
algorithm for both problems for a large class of objects, significantly
improving the best known approximation guarantees.
