$\newcommand{\eps}{\varepsilon}$ 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，一个适用于输入图任意诱导子图的$(1-\eps)$-近似独立集算法，可通过多项式时间归约，用于提供$(1+\eps)$-近似顶点覆盖算法。这一基本观察此前已有记载，参见[BHR11]。由此，我们得到：无加权伪圆盘的顶点覆盖问题的PTAS、平面无加权轴对齐矩形的顶点覆盖问题的QQPTAS，以及平面加权多边形的MWVC问题的QPTAS。据我们所知，这些结果均为首次提出。