$\newcommand{\eps}{\varepsilon}$ We observe that a $(1-\eps)$-approximation algorithm to Independent Set, that works for any induced subgraph of the input graph, can be used, via a polynomial time reduction, to provide a $(1+\eps)$-approximation to Vertex Cover. This basic observation was made before, see [BHR11]. As a consequence, we get a PTAS for VC for unweighted pseudo-disks, QQPTAS for VC for unweighted axis-aligned rectangles in the plane, and QPTAS for MWVC for weighted polygons in the plane. To the best of our knowledge all these results are new.

翻译：$\newcommand{\eps}{\varepsilon}$ 我们观察到，若存在一个针对输入图任意导出子图的 $(1-\eps)$ 近似算法求解独立集问题，则可通过多项式时间归约，得到一个求解顶点覆盖问题的 $(1+\eps)$ 近似算法。这一基本观察前人已有提及，参见[BHR11]。由此，我们得到了针对无权伪圆盘的顶点覆盖问题的PTAS，针对平面内无权轴对齐矩形的顶点覆盖问题的QQPTAS，以及针对平面内加权多边形的最大权顶点覆盖问题（MWVC）的QPTAS。据我们所知，所有这些结果均为首次提出。