基于最优分离器的几何集覆盖问题的QPTAS
QPTAS for Geometric Set-Cover Problems via Optimal Separators
摘要：
加权几何集合覆盖问题在几种几何和非几何设置中自然出现（例如Bansal-Pruhs（FOCS 2010）的突破将广泛的机器调度问题减少到加权几何集合覆盖）。二十多年的研究已经成功地解决了大多数几何集合覆盖问题的$(1+ε)$ -可逼近性状态，除了四个基本场景仍然缺乏。一种是平面中的加权圆盘，在一系列论文之后，Varadarajan（STOC 2010）提出了一种巧妙的\emph{quasi-sampling}技术，该技术与Chan \etal ~（SODA 2012）的改进一起产生了$O(1)$ -近似算法。甚至对于未加权的情况，对于称为伪圆盘的基本类对象（包括圆盘、单位高度矩形、凸集的平移等）的PTAS也是有效的。目前尚不清楚。另一个基本情况是$\Re^3$中的加权半空间，目前缺乏PTAS。在本文中，我们提出了一个QPTAS所有这些剩余的问题。我们的结果是基于Adamaszek-Wiese（FOCS 2013，SODA 2014）的分离器框架，他最近获得了多边形区域的加权独立集的QPTAS。这排除了这些问题是APX困难的可能性，假设是$\textbf{NP} \nsubseteq \textbf{DTIME}(2^{polylog(n)})$。结合Chan-Grant最近的工作（CGTA 2014），这解决了所有自然几何集覆盖问题的APX-硬度状态。
英文摘要：
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 $(1+ε)$-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.
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