12,641 research outputs found
Towards Tight Bounds for the Streaming Set Cover Problem
We consider the classic Set Cover problem in the data stream model. For
elements and sets () we give a -pass algorithm with a
strongly sub-linear space and logarithmic
approximation factor. This yields a significant improvement over the earlier
algorithm of Demaine et al. [DIMV14] that uses exponentially larger number of
passes. We complement this result by showing that the tradeoff between the
number of passes and space exhibited by our algorithm is tight, at least when
the approximation factor is equal to . Specifically, we show that any
algorithm that computes set cover exactly using passes
must use space in the regime of .
Furthermore, we consider the problem in the geometric setting where the
elements are points in and sets are either discs, axis-parallel
rectangles, or fat triangles in the plane, and show that our algorithm (with a
slight modification) uses the optimal space to find a
logarithmic approximation in passes.
Finally, we show that any randomized one-pass algorithm that distinguishes
between covers of size 2 and 3 must use a linear (i.e., ) amount of
space. This is the first result showing that a randomized, approximate
algorithm cannot achieve a space bound that is sublinear in the input size.
This indicates that using multiple passes might be necessary in order to
achieve sub-linear space bounds for this problem while guaranteeing small
approximation factors.Comment: A preliminary version of this paper is to appear in PODS 201
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
Approximating Smallest Containers for Packing Three-dimensional Convex Objects
We investigate the problem of computing a minimal-volume container for the
non-overlapping packing of a given set of three-dimensional convex objects.
Already the simplest versions of the problem are NP-hard so that we cannot
expect to find exact polynomial time algorithms. We give constant ratio
approximation algorithms for packing axis-parallel (rectangular) cuboids under
translation into an axis-parallel (rectangular) cuboid as container, for
cuboids under rigid motions into an axis-parallel cuboid or into an arbitrary
convex container, and for packing convex polyhedra under rigid motions into an
axis-parallel cuboid or arbitrary convex container. This work gives the first
approximability results for the computation of minimal volume containers for
the objects described
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