155 research outputs found
Computational Geometry Column 41
The recent result that n congruent balls in R^d have at most 4 distinct
geometric permutations is described.Comment: To appear in SIGACT News and in Internat. J. Comput. Geom. App
Heterochromatic Higher Order Transversals for Convex Sets
In this short paper, we show that if be a collection of families compact -fat convex sets in
and if every heterochromatic sequence with respect to
contains convex sets
that can be pierced by a -flat then there exists a family
from the collection that can be pierced by finitely many -flats.
Additionally, we show that if be a collection of families of compact convex sets in
where each is a family of closed balls (axis
parallel boxes) in and every heterochromatic sequence with
respect to contains
intersecting closed balls (boxes) then there exists a family
from the collection that can be pierced by a finite number of points from
. To complement the above results, we also establish some
impossibility of proving similar results for other more general families of
convex sets.
Our results are a generalization of -Theorem for
-transversals of convex sets by Keller and Perles (Symposium on
Computational Geometry 2022), and can also be seen as a colorful infinite
variant of -Theorems of Alon and Klietman (Advances in Mathematics
1992), and Alon and Kalai (Discrete & Computational Geometry 1995).Comment: 16 pages and 5 figures. Section 3 rewritte
QPTAS for Weighted Geometric Set Cover on Pseudodisks and Halfspaces
International audienceWeighted geometric set-cover problems arise naturally in several geometric and non-geometric settings (e.g. the breakthrough of Bansal and 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 + 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 (STOC 2010) presented a clever quasi-sampling technique, which together with improvements by Chan et al. (SODA 2012), yielded an O(1)-approximation algorithm. Even for the unweighted case, a PTAS for a fundamental class of objects called pseudodisks (which includes half-spaces, 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 QPTAS for all of these remaining problems. Our results are based on the separator framework of Adamaszek and 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 NP DTIME(2 polylog(n)). Together with the recent work of Chan and Grant (CGTA 2014), this settles the APX-hardness status for all natural geometric set-cover problems
Packing and covering with balls on Busemann surfaces
In this note we prove that for any compact subset of a Busemann surface
(in particular, for any simple polygon with geodesic metric)
and any positive number , the minimum number of closed balls of radius
with centers at and covering the set is at most 19
times the maximum number of disjoint closed balls of radius centered
at points of : , where and
are the covering and the packing numbers of by -balls.Comment: 27 page
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