Capacitated Discrete Unit Disk Cover

  • Pawan K. Mishra
  • Sangram K. Jena
  • Gautam K. DasEmail author
  • S. V. Rao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11355)


Consider a capacitated version of the discrete unit disk cover problem as follows: consider a set \(P= \{p_1,p_2, \cdots ,p_n\}\) of n customers and a set \(Q=\{q_1,q_2, \cdots ,q_m\}\) of m service centers. A service center can provide service to at most \(\alpha ( \in \mathbb {N})\) number of customers. Each \(q_i \in Q\) \((i=1,2, \cdots ,m)\) has a preassigned set of customers to which it can provide service. The objective of the capacitated covering problem is to provide service to each customer in P by at least one service center in Q. In this paper, we consider the geometric version of the capacitated covering problem, where the set of customers and set of service centers are two point sets in the Euclidean plane. A service center can provide service to a customer if their Euclidean distance is less than or equal to 1. We call this problem as \((\alpha , P, Q)\)-covering problem. For the \((\alpha , P, Q)\)-covering problem, we propose an \(O(\alpha mn(m+n))\) time algorithm to check feasible solution for a given instance. We also prove that the \((\alpha , P, Q)\)-covering problem is NP-complete for \(\alpha \ge 3\) and it admits a PTAS.


Geometric covering NP-complete PTAS 


  1. 1.
    Ambühl, C., Erlebach, T., Mihalák, M., Nunkesser, M.: Constant-factor approximation for minimum-weight (connected) dominating sets in unit disk graphs. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds.) APPROX/RANDOM -2006. LNCS, vol. 4110, pp. 3–14. Springer, Heidelberg (2006). Scholar
  2. 2.
    Basappa, M., Acharyya, R., Das, G.K.: Unit disk cover problem in 2D. J. Discrete Algorithms 33, 193–201 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Biedl, T., Kant, G.: A better heuristic for orthogonal graph drawings. Comput. Geom. 9(3), 159–180 (1998)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. Discrete Comput. Geom. 14(4), 463–479 (1995)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Călinescu, G., Mandoiu, I.I., Wan, P.J., Zelikovsky, A.Z.: Selecting forwarding neighbors in wireless ad hoc networks. Mobile Netw. Appl. 9(2), 101–111 (2004)CrossRefGoogle Scholar
  6. 6.
    Carmi, P., Katz, M.J., Lev-Tov, N.: Covering points by unit disks of fixed location. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 644–655. Springer, Heidelberg (2007). Scholar
  7. 7.
    Claude, F., et al.: An improved line-separable algorithm for discrete unit disk cover. Discrete Math. Algorithms Appl. 2(01), 77–87 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Das, G.K., Fraser, R., López-Ortiz, A., Nickerson, B.G.: On the discrete unit disk cover problem. Int. J. Comput. Geom. Appl. 22(05), 407–419 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Federickson, G.N.: Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Comput. 16(6), 1004–1022 (1987)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Feige, U.: A threshold of ln n for approximating set cover. J. ACM (JACM) 45(4), 634–652 (1998)CrossRefGoogle Scholar
  11. 11.
    Fraser, R., López-Ortiz, A.: The within-strip discrete unit disk cover problem. Theor. Comput. Sci. 674, 99–115 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)zbMATHGoogle Scholar
  13. 13.
    Haussler, D., Welzl, E.: \(\epsilon \)-nets and simplex range queries. Discrete & Computational Geometry 2(2), 127–151 (1987)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kleinberg, J., Tardos, E.: Algorithm Design. Addison-Wesley Longman Publishing Co. Inc, Boston (2005)Google Scholar
  15. 15.
    Mustafa, N.H., Ray, S.: Improved results on geometric hitting set problems. Discrete Comput. Geom. 44(4), 883–895 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mustafa, N.H., Ray, S.: PTAS for geometric hitting set problems via local search. In: Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry, pp. 17–22. ACM (2009)Google Scholar
  17. 17.
    Valiant, L.G.: Universality considerations in VLSI circuits. IEEE Trans. Comput. 100(2), 135–140 (1981)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Pawan K. Mishra
    • 1
  • Sangram K. Jena
    • 1
  • Gautam K. Das
    • 1
    Email author
  • S. V. Rao
    • 1
  1. 1.Indian Institute of TechnologyGuwahatiIndia

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