4,455 research outputs found
Minimum Convex Partitions and Maximum Empty Polytopes
Let be a set of points in . A Steiner convex partition
is a tiling of with empty convex bodies. For every integer ,
we show that admits a Steiner convex partition with at most tiles. This bound is the best possible for points in general
position in the plane, and it is best possible apart from constant factors in
every fixed dimension . We also give the first constant-factor
approximation algorithm for computing a minimum Steiner convex partition of a
planar point set in general position. Establishing a tight lower bound for the
maximum volume of a tile in a Steiner convex partition of any points in the
unit cube is equivalent to a famous problem of Danzer and Rogers. It is
conjectured that the volume of the largest tile is .
Here we give a -approximation algorithm for computing the
maximum volume of an empty convex body amidst given points in the
-dimensional unit box .Comment: 16 pages, 4 figures; revised write-up with some running times
improve
Querying for the Largest Empty Geometric Object in a Desired Location
We study new types of geometric query problems defined as follows: given a
geometric set , preprocess it such that given a query point , the
location of the largest circle that does not contain any member of , but
contains can be reported efficiently. The geometric sets we consider for
are boundaries of convex and simple polygons, and point sets. While we
primarily focus on circles as the desired shape, we also briefly discuss empty
rectangles in the context of point sets.Comment: This version is a significant update of our earlier submission
arXiv:1004.0558v1. Apart from new variants studied in Sections 3 and 4, the
results have been improved in Section 5.Please note that the change in title
and abstract indicate that we have expanded the scope of the problems we
stud
Covering Points by Disjoint Boxes with Outliers
For a set of n points in the plane, we consider the axis--aligned (p,k)-Box
Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together
contain n-k points. In this paper, we consider the boxes to be either squares
or rectangles, and we want to minimize the area of the largest box. For general
p we show that the problem is NP-hard for both squares and rectangles. For a
small, fixed number p, we give algorithms that find the solution in the
following running times:
For squares we have O(n+k log k) time for p=1, and O(n log n+k^p log^p k time
for p = 2,3. For rectangles we get O(n + k^3) for p = 1 and O(n log n+k^{2+p}
log^{p-1} k) time for p = 2,3.
In all cases, our algorithms use O(n) space.Comment: updated version: - changed problem from 'cover exactly n-k points' to
'cover at least n-k points' to avoid having non-feasible solutions. Results
are unchanged. - added Proof to Lemma 11, clarified some sections - corrected
typos and small errors - updated affiliations of two author
Approximation Schemes for Maximum Weight Independent Set of Rectangles
In the Maximum Weight Independent Set of Rectangles (MWISR) problem we are
given a set of n axis-parallel rectangles in the 2D-plane, and the goal is to
select a maximum weight subset of pairwise non-overlapping rectangles. Due to
many applications, e.g. in data mining, map labeling and admission control, the
problem has received a lot of attention by various research communities. We
present the first (1+epsilon)-approximation algorithm for the MWISR problem
with quasi-polynomial running time 2^{poly(log n/epsilon)}. In contrast, the
best known polynomial time approximation algorithms for the problem achieve
superconstant approximation ratios of O(log log n) (unweighted case) and O(log
n / log log n) (weighted case).
Key to our results is a new geometric dynamic program which recursively
subdivides the plane into polygons of bounded complexity. We provide the
technical tools that are needed to analyze its performance. In particular, we
present a method of partitioning the plane into small and simple areas such
that the rectangles of an optimal solution are intersected in a very controlled
manner. Together with a novel application of the weighted planar graph
separator theorem due to Arora et al. this allows us to upper bound our
approximation ratio by (1+epsilon).
Our dynamic program is very general and we believe that it will be useful for
other settings. In particular, we show that, when parametrized properly, it
provides a polynomial time (1+epsilon)-approximation for the special case of
the MWISR problem when each rectangle is relatively large in at least one
dimension. Key to this analysis is a method to tile the plane in order to
approximately describe the topology of these rectangles in an optimal solution.
This technique might be a useful insight to design better polynomial time
approximation algorithms or even a PTAS for the MWISR problem
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