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 $\mathsf{APX}$-hard for object sets consisting of axis-parallel rectangles, ellipses, α-fat objects of constant description complexity, and convex polygons. On the other hand, $\mathsf{PTAS}$s (polynomial time approximation schemes) are known for object sets consisting of disks or unit squares. Surprisingly, a $\mathsf{PTAS}$ was unknown even for arbitrary squares. For both problems obtaining a $\mathsf{PTAS}$ 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 $\mathsf{PTAS}$ 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.

在本文中，我们研究了两个经典的优化问题：最小几何支配集和集合覆盖。在支配集问题中，对于平面中给定的一组对象作为输入，目标是选择最少数量的输入对象，使得每个输入对象都由所选对象集支配。在这里，如果一个对象的交集是非空的，我们就说一个对象受另一个对象支配。对于第二个问题，对于平面中给定的一组点和对象，目标是选择最少数量的对象来覆盖所有的点。这是集合覆盖问题的一个特殊版本。

这两个问题都得到了很好的研究，受到输入对象的各种限制。这些问题是$\mathsf{亚太证券交易所}$-hard 适用于由轴平行的矩形、椭圆、具有恒定描述复杂度的α -fat 对象和凸多边形组成的对象集。另一方面，$\mathsf{PTAS}$s（多项式时间近似方案）以由圆盘或单位正方形组成的对象集而闻名。令人惊讶的是，一个$\mathsf{PTAS}$即使对于任意正方形也是未知的。对于这两个问题，获得$\mathsf{PTAS}$对一大类对象保持开放。

对于支配集问题，我们证明了一种流行的局部搜索算法会导致$(\mathrm{1个}+\epsilon )$凸对象（包括任意正方形、k-正多边形、凸集的平移和缩放副本等）的同位素族的近似值$n^{欧(\mathrm{1个}/{\epsilon }^{\mathrm{2个}})}$时间。另一方面，同样的方法导致$\mathsf{PTAS}$针对对象为凸伪圆盘（包括圆盘、单位高矩形、类位凸对象等）时的几何覆盖问题。因此，我们为一大类对象的两个问题获得了一个易于实现的近似算法，显着提高了最知名的近似保证。