Weighted geometric set-cover problems arise naturally in several geometric and non-geometric settings (e.g. the breakthrough of Bansal-Pruhs (FOCS 2010) reduces a wide class of machine scheduling problems to weighted geometric set-cover). More than two decades of research has succeeded in settling the $(1+\epsilon)$-approximability status for most geometric set-cover problems, except for four basic scenarios which are still lacking. One is that of weighted disks in the plane for which, after a series of papers, Varadarajan (STOC 2010) presented a clever \emph{quasi-sampling} technique, which together with improvements by Chan \etal~(SODA 2012), yielded a $O(1)$-approximation algorithm. Even for the unweighted case, a PTAS for a fundamental class of objects called pseudodisks (which includes disks, unit-height rectangles, translates of convex sets etc.) is currently unknown. Another fundamental case is weighted halfspaces in $\Re^3$, for which a PTAS is currently lacking. In this paper, we present a QPTAS for all of these remaining problems. Our results are based on the separator framework of Adamaszek-Wiese (FOCS 2013, SODA 2014), who recently obtained a QPTAS for weighted independent set of polygonal regions. This rules out the possibility that these problems are APX-hard, assuming $\textbf{NP} \nsubseteq \textbf{DTIME}(2^{polylog(n)})$. Together with the recent work of Chan-Grant (CGTA 2014), this settles the APX-hardness status for all natural geometric set-cover problems.
翻译:在数个几何和非几何设置中自然出现加权地谱覆盖问题(例如,Bansal-Pruhs的突破(FOCS,2010年)将机器调度问题降低到加权几何套套套套。超过20年的研究成功地解决了大部分几何套套套套问题的$(1 ⁇ epsilon)$-兼容性状态,但四个基本假设仍然缺乏。其中一个是平面上的加权磁盘(STOC,2010年)在一系列论文之后,Varaadarajan(STOC,2010年)展示了智能的 emph{quas-scamping} 技术,这与Chan\etal~(SODAD,2012年)的改进一起,产生了一个美元($(1)美元)的适应性算法算法。即使是未加权的,对于所谓的几类基本天体(包括磁盘、单位-十八正对等方位矩、将正对等式的离心机数据集等),目前还不清楚。另一个基本案例是 Q\\Q-RexAS-real-al-alalalal a fal rodeal rualalalalalalalal,我们目前缺乏的Slade Slax 。