On Covering Problems of Rado

@article{Bereg2009OnCP,
  title={On Covering Problems of Rado},
  author={Sergey Bereg and Adrian Dumitrescu and Minghui Jiang},
  journal={Algorithmica},
  year={2009},
  volume={57},
  pages={538-561}
}
T. Rado conjectured in 1928 that if ℱ is a finite set of axis-parallel squares in the plane, then there exists an independent subset ℐ⊆ℱ of pairwise disjoint squares, such that ℐ covers at least 1/4 of the area covered by ℱ. He also showed that the greedy algorithm (repeatedly choose the largest square disjoint from those previously selected) finds an independent set of area at least 1/9 of the area covered by ℱ. The analogous question for other shapes and many similar problems have been… Expand
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References

SHOWING 1-10 OF 39 REFERENCES
Optimal Packing and Covering in the Plane are NP-Complete
TLDR
This paper proves that even severely restricted instances of packing and covering problems remain NP-hard in two or more dimensions, and helps to fill the gap by showing that some very constrained intersection graph problems in two dimensions are not very constrained. Expand
A SOLUTION TO A PROBLEM OF
We announce a solution to a multiplicity problem for nests posed by J. R. Ringrose approximately twenty years ago. This also answers a question posed by R. V. Kadison and I. M. Singer, andExpand
Polynomial-Time Approximation Schemes for Geometric Intersection Graphs
TLDR
These are the first known PTASs for $\mathcal{NP}$-hard optimization problems on disk graphs based on a novel recursive subdivision of the plane that allows applying a shifting strategy on different levels simultaneously, so that a dynamic programming approach becomes feasible. Expand
Approximation schemes for covering and packing problems in image processing and VLSI
TLDR
The unified technique that is introduced here, referred to as the shifting strategy, is applicable to numerous geometric covering and packing problems and how it varies with problem parameters is illustrated. Expand
Research problems in discrete geometry
TLDR
This chapter discusses 100 Research Problems in Discrete Geometry from the Facsimile edition of the World Classics in Mathematics Series, vol. Expand
Unsolved Problems In Geometry
TLDR
A monograph on geometry, each section in the book describes a problem or a group of related problems, capable of generalization of variation in many directions. Expand
Maximum Area Independent Sets in Disk Intersection Graphs
TLDR
Quantitative bounds on the maximum total area of an independent set relative to the union area and Practical constant-ratio approximation algorithms for finding anindependent set with a large total area relative toThe union area are obtained. Expand
Computational geometry: an introduction
TLDR
This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry. Expand
Combinatorial geometry
  • J. Pach, P. Agarwal
  • Mathematics, Computer Science
  • Wiley-Interscience series in discrete mathematics and optimization
  • 1995
\indent This beautiful discipline emerged from number theory after the fruitful observation made by Minkowski (1896) that many important results in diophantine approximation (and in some otherExpand
CRC standard mathematical tables and formulae; 30th edition
Numbers and Elementary Mathematics Proofs without words Constants Special numbers Number theory Series and products Algebra Elementary algebra Polynomials Vector algebra Linear and matrix algebraExpand
...
1
2
3
4
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