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On Covering Problems of Rado

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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.

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Authors and Affiliations

  1. Department of Computer Science, University of Texas at Dallas, Box 830688, Richardson, TX, 75083, USA

    Sergey Bereg

  2. Department of Computer Science, University of Wisconsin–Milwaukee, Milwaukee, WI, 53201-0784, USA

    Adrian Dumitrescu

  3. Department of Computer Science, Utah State University, Logan, UT, 84322-4205, USA

    Minghui Jiang

Authors
  1. Sergey Bereg
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  2. Adrian Dumitrescu
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  3. Minghui Jiang
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Corresponding author

Correspondence to Minghui Jiang.

Additional information

A. Dumitrescu was partially supported by NSF CAREER grant CCF-0444188.

M. Jiang was partially supported by NSF grant DBI-0743670.

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Bereg, S., Dumitrescu, A. & Jiang, M. On Covering Problems of Rado. Algorithmica 57, 538–561 (2010). https://doi.org/10.1007/s00453-009-9298-z

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