Algorithmica

, Volume 57, Issue 3, pp 538–561 | Cite as

On Covering Problems of Rado

Article

Abstract

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 considered by R. Rado in his three papers (in Proc. Lond. Math. Soc. 51:232–264, 1949; 53:243–267, 1951; and J. Lond. Math. Soc. 42:127–130, 1968) on this subject. After 45 years, Ajtai (in Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 21:61–63, 1973) came up with a surprising example disproving T. Rado’s conjecture. We revisit Rado’s problem and present improved upper and lower bounds for squares, disks, convex bodies, centrally symmetric convex bodies, and others, as well as algorithmic solutions to these variants of the problem.

Keywords

Discrete and computational geometry Approximation algorithms 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Sergey Bereg
    • 1
  • Adrian Dumitrescu
    • 2
  • Minghui Jiang
    • 3
  1. 1.Department of Computer ScienceUniversity of Texas at DallasRichardsonUSA
  2. 2.Department of Computer ScienceUniversity of Wisconsin–MilwaukeeMilwaukeeUSA
  3. 3.Department of Computer ScienceUtah State UniversityLoganUSA

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