  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.
