A Polynomial-Time Approximation Scheme for the Geometric Unique Coverage Problem on Unit Squares

  • Takehiro Ito
  • Shin-Ichi Nakano
  • Yoshio Okamoto
  • Yota Otachi
  • Ryuhei Uehara
  • Takeaki Uno
  • Yushi Uno
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)


We give a polynomial-time approximation scheme for the unique unit-square coverage problem: given a set of points and a set of axis-parallel unit squares, both in the plane, we wish to find a subset of squares that maximizes the number of points contained in exactly one square in the subset. Erlebach and van Leeuwen (2008) introduced this problem as the geometric version of the unique coverage problem, and the best approximation ratio by van Leeuwen (2009) before our work was 2. Our scheme can be generalized to the budgeted unique unit-square coverage problem, in which each point has a profit, each square has a cost, and we wish to maximize the total profit of the uniquely covered points under the condition that the total cost is at most a given bound.


Polynomial Time Unit Disk Knapsack Problem Coverage Problem Lower Envelope 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Baker, B.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41, 153–180 (1994)zbMATHCrossRefGoogle Scholar
  2. 2.
    Bazgan, C.: Schémas d’approximation et complexité paramétrée. Rapport de DEA, Université Paris Sud (1995)Google Scholar
  3. 3.
    Cesati, M., Trevisan, L.: On the efficiency of polynomial time approximation schemes. Information Processing Letters 64, 165–171 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chandra, A.K., Hirschberg, D.S., Wong, C.K.: Approximate algorithms for some generalized knapsack problems. Theoretical Computer Science 3, 293–304 (1976)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Mathematics 86, 165–177 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Demaine, E.D., Hajiaghayi, M.T., Feige, U., Salavatipour, M.R.: Combination can be hard: approximability of the unique coverage problem. SIAM J. on Computing 38, 1464–1483 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Dumitrescu, A., Jiang, M.: Dispersion in unit disks. In: Proc. STACS 2010, pp. 299–310 (2010)Google Scholar
  8. 8.
    Erlebach, T., van Leeuwen, E.J.: Approximating geometric coverage problems. In: Proc. SODA 2008, pp. 1267–1276 (2008)Google Scholar
  9. 9.
    Fiala, J., Kratochvíl, J., Proskurowski, A.: Systems of distant representatives. Discrete Applied Mathematics 145, 306–316 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Guruswami, V., Trevisan, L.: The Complexity of Making Unique Choices: Approximating 1-in-k SAT. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 99–110. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32, 130–136 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Hunt III, H.B., Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: NC-approximation schemes for NP- and PSPACE-hard problems for geometric graphs. J. Algorithms 26, 238–274 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Huson, M.L., Sen, A.: Broadcast scheduling algorithms for radio networks. In: Proc. IEEE MILCOM 1995, pp. 647–651 (1995)Google Scholar
  14. 14.
    Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer (2004)Google Scholar
  15. 15.
    Misra, N., Raman, V., Saurabh, S., Sikdar, S.: The Budgeted Unique Coverage Problem and Color-Coding. In: Frid, A., Morozov, A., Rybalchenko, A., Wagner, K.W. (eds.) CSR 2009. LNCS, vol. 5675, pp. 310–321. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Moser, H., Raman, V., Sikdar, S.: The Parameterized Complexity of the Unique Coverage Problem. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 621–631. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley (1994)Google Scholar
  18. 18.
    van Leeuwen, E.J.: Optimization and approximation on systems of geometric objects. Ph.D. Thesis, University of Amsterdam (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Takehiro Ito
    • 1
  • Shin-Ichi Nakano
    • 2
  • Yoshio Okamoto
    • 3
  • Yota Otachi
    • 4
  • Ryuhei Uehara
    • 4
  • Takeaki Uno
    • 5
  • Yushi Uno
    • 6
  1. 1.Graduate School of Information SciencesTohoku UniversityJapan
  2. 2.Department of Computer ScienceGunma UniversityJapan
  3. 3.Department of Communication Engineering and InformaticsUniversity of Electro-CommunicationsJapan
  4. 4.School of Information ScienceJAISTJapan
  5. 5.Principles of Informatics Research DivisionNational Institute of InformaticsJapan
  6. 6.Graduate School of ScienceOsaka Prefecture UniversityJapan

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