Geometric dominating-set and set-cover via local-search

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, we say that one object is dominated by another if their intersection is nonempty. For the second problem, for a given set of points and objects in the plane, the objective is to choose a minimum number of objects to cover all the points. This is a particular version of the set-cover problem.

Both problems have been well-studied, subject to various restrictions on the input objects. These problems are $\mathrm{APX}$-hard for object sets consisting of axis-parallel rectangles, ellipses, α-fat objects of constant description complexity, and convex polygons. On the other hand, $\mathrm{PTAS}$s (polynomial time approximation schemes) are known for object sets consisting of disks or unit squares. Surprisingly, a $\mathrm{PTAS}$ was unknown even for arbitrary squares. For both problems obtaining a $\mathrm{PTAS}$ remains open for a large class of objects.

For the dominating-set problem, we prove that a popular local-search algorithm leads to a $(1+\epsilon )$ approximation for a family of homothets of a convex object (which includes arbitrary squares, k-regular polygons, translated and scaled copies of a convex set, etc.) in $n^{O(1/{\epsilon }^2)}$ time. On the other hand, the same approach leads to a $\mathrm{PTAS}$ for the geometric covering problem when the objects are convex pseudodisks (which include disks, unit height rectangles, homothetic convex objects, etc.). Consequently, we obtain an easy-to-implement approximation algorithm for both problems for a large class of objects, significantly improving the best-known approximation guarantees.