Quasi-Polynomial Time Approximation Scheme for Weighted Geometric Set Cover on Pseudodisks and Halfspaces

Weighted geometric set-cover problems arise naturally in several geometric and nongeometric settings (e.g., the breakthrough of Bansal and Pruhs [Proceedings of FOCS, 2010, pp. 407-414] 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 some basic scenarios which are still lacking. One is that of weighted disks in the plane for which, after a series of papers, Varadarajan [Proceedings of STOC'10, 2010, pp. 641-648] presented a clever quasi-sampling technique, which together with improvements by Chan et al. [Proceedings of SODA, 2012, pp. 1576-1585], yielded an O(1)-approximation algorithm. Even for the unweighted case, a polynomial time approximation scheme (PTAS) for a fundamental class of objects called pseudodisks (which includes halfspaces, disks, unit-height rectangles, translates of convex sets, etc.) is currently unknown. Another fundamental case is weighted halfspaces in R-3, for which a PTAS is currently lacking. In this paper, we present a quasi PTAS (QPTAS) for all these remaining problems. Our results are based on the separator framework of Adamaszek and Wiese [Proceedings of FOCS, 2013, pp. 400-409; Proceedings of SODA, 2014, pp. 645-656], who recently obtained a QPTAS for a weighted independent set of polygonal regions. This rules out the possibility that these problems are APX-hard, assuming NP not subset of DTIME(2(polylog(n))). Together with the recent work of Chan and Grant [Comput. Geom., 47 (2014), pp. 112-124], this settles the APX-hardness status for all natural geometric set-cover problems.