On Covering Problems of Rado

  • Sergey Bereg
  • Adrian Dumitrescu
  • Minghui Jiang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5124)


T. Rado conjectured in 1928 that if \({\mathcal S}\) is a finite set of axis-parallel squares in the plane, then there exists an independent subset \({\mathcal I}\subseteq\) \({\mathcal S}\) of pairwise disjoint squares, such that \({\mathcal I}\) covers at least 1/4 of the area covered by \({\mathcal S}\). 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 \({\mathcal S}\). The analogous question for other shapes and many similar problems have been considered by R. Rado in his three papers (1949, 1951 and 1968) on this subject. After 45 years (in 1973), Ajtai 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 sets, centrally symmetric convex sets, and others, as well as algorithmic solutions to these variants of the problem.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ajtai, M.: The solution of a problem of T. Rado. Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Math. Astr. et Phys. 21, 61–63 (1973)MATHMathSciNetGoogle Scholar
  2. 2.
    Bereg, S., Dumitrescu, A., Jiang, M.: Maximum area independent set in disk intersection graphs. International Journal of Computational Geometry & Applications (to appear)Google Scholar
  3. 3.
    Chan, T.: Polynomial-time approximation schemes for packing and piercing fat objects. Journal of Algorithms 46, 178–189 (2003)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Croft, H.T., Falconer, K.J., Guy, R.K.: Unsolved Problems in Geometry. Springer, New York (1991)MATHGoogle Scholar
  5. 5.
    Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time approximation schemes for geometric intersection graphs. SIAM Journal on Computing 34, 1302–1323 (2005)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Fowler, R.J., Paterson, M.S., Tanimoto, S.L.: Optimal packing and covering in the plane are NP-complete. Information Processing Letters 12, 133–137 (1981)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. Journal of ACM 32, 130–136 (1985)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Norlander, G.: A covering problem. Nordisk Mat. Tidskr. 6, 29–31 (1958)MathSciNetGoogle Scholar
  9. 9.
    Rado, R.: Some covering theorems (I). Proceedings of the London Mathematical Society 51, 241–264 (1949)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Rado, R.: Some covering theorems (II). Proceedings of the London Mathematical Society 53, 243–267 (1951)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Rado, R.: Some covering theorems (III). Journal of the London Mathematical Society 42, 127–130 (1968)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Rado, T.: Sur un problème relatif à un théorème de Vitali. Fund. Math. 11, 228–229 (1928)Google Scholar
  13. 13.
    Sokolin, A.: Concerning a problem of Rado. C.R. Acad. Sci. U.R.S.S (N.S.) 26, 871–872 (1940)MathSciNetGoogle Scholar
  14. 14.
    Yaglom, I.M., Boltyanskiĭ, V. G.: Convex Figures. Holt, Rinehart and Winston, New York (1961)Google Scholar
  15. 15.
    Zalgaller, V.A.: Remarks on a problem of Rado. Matem. Prosveskcheric 5, 141–148 (1960)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

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–MilwaukeeUSA
  3. 3.Department of Computer ScienceUtah State UniversityLoganUSA

Personalised recommendations