Advertisement

On a Class of Covering Problems with Variable Capacities in Wireless Networks

  • Selim Akl
  • Robert Benkoczi
  • Daya Ram Gaur
  • Hossam Hassanein
  • Shahadat Hossain
  • Mark Thom
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8344)

Abstract

We consider the problem of allocating clients to base stations in wireless networks. Two design decisions are the location of the base stations, and the power levels of the base stations. We model the interference due to the increased power usage resulting in greater serving radius, as capacities that are non-increasing with respect to the covering radius. We consider three models. In the first model the location of the base stations and the clients are fixed, and the problem is to determine the serving radius for each base station so as to serve a set of clients with maximum total profit subject to the capacity constraints of the base stations. In the second model, each client has an associated demand in addition to its profit. A fixed number of facilities have to be opened from a candidate set of locations. The goal is to serve clients so as to maximize the profit subject to the capacity constraints. In the third model the location and the serving radius of the base stations are to be determined. There are costs associated with opening the base stations, and the goal is to open a set of base stations of minimum total cost so as to serve the entire client demand subject to the capacity constraints at the base stations. We show that for the first model the problem is NP-complete even when there are only two choices for the serving radius, and the capacities are 1, 2. For the second model we give a 1/2-ε approximation algorithm. For the third model we give a column generation procedure for solving the standard linear programming model, and a randomized rounding procedure. We establish the efficacy of the column generation based rounding scheme on randomly generated instances.

Keywords

Greedy Algorithm Facility Location Capacity Constraint Knapsack Problem Covering Problem 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aardal, K.: Capacitated facility location: Separation algorithms and computational experience. Math. Program. 81, 149–175 (1998)zbMATHMathSciNetGoogle Scholar
  2. 2.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, ch. 5, 3rd edn. Springer (2008) ISBN: 978-3-540-77973-5Google Scholar
  3. 3.
    Berman, O., Drezner, Z., Krass, D.: Generalized coverage: New developments in covering location models. Computers & Operations Research 37(10), 1675–1687 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Catrein, D., Imhof, L.A., Mathar, R.: Power control, capacity, and duality of uplink and downlink in cellular CDMA systems. IEEE Transactions on Communications 52(10), 1777–1785 (2004)CrossRefGoogle Scholar
  5. 5.
    Chudak, F.A., Williamson, D.P.: Improved approximation algorithms for capacitated facility location problems. Math. Program. 102(2), 207–222 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Fisher, M.L., Nemhauser, G.L., Wolsey, L.A.: An analysis of the approximations for maximizing submodular set functions II. Mathematical Programming Study 8, 73–87 (1978)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Fleischer, L., Goemans, M.X., Mirrokni, V.S., Sviridenko, M.: Tight approximation algorithms for maximum general assignment problem. In: SODA, pp. 611–620 (2006)Google Scholar
  8. 8.
    Hanly, S.: Congestion measures in DS-CDMA networks. IEEE Transactions on Communications 47(3), 426–437 (1999)CrossRefGoogle Scholar
  9. 9.
    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32(1), 130–136 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Holma, H., Toskala, A.: WCDMA for UMTS: HSPA Evolution and LTE, 4th edn. Wiley (2007) ISBN: 978-0470319338Google Scholar
  11. 11.
    Ibarra, O.H., Kim, C.E.: Fast approximation algorithms for the knapsack and sum of subsets problems. J. ACM 22, 463–468 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Martello, S., Toth, P.: Knapsack Problems: Algorithms and Computer Implementations, Revised edn. Wiley (1990) ISBN: 978-0471924203Google Scholar
  13. 13.
    Mulvey, J.M., Beck, M.P.: Solving capacitated clustering problems. European Journal of Operational Research 18(3), 339–348 (1984)CrossRefzbMATHGoogle Scholar
  14. 14.
    Radwan, A., Hassanein, H.: Capacity enhancement in CDMA cellular networks using multi-hop communication. In: Proceedings of the 11th IEEE Symposium on Computers and Communications, June 26-29, pp. 832–837. IEEE (2006)Google Scholar
  15. 15.
    Schrijver, A. Combinatorial Optimization, first ed., vol. A, part II. Springer, ch. 21, pp. 337–377. ISBN: 978-3540443896 (2003)Google Scholar
  16. 16.
    Tam, Y.H., Hassanein, H.S., Akl, S.G., Benkoczi, R.: Optimal multi-hop cellular architecture for wireless communications. In: Proceedings of the 31st IEEE Conference on Local Computer Networks, pp. 738–745 (November 2006)Google Scholar
  17. 17.
    Vondrak, J.: Optimal approximation for the submodular welfare problem in value oracle model. In: STOC, pp. 67–74 (2008)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Selim Akl
    • 1
  • Robert Benkoczi
    • 2
  • Daya Ram Gaur
    • 2
  • Hossam Hassanein
    • 1
  • Shahadat Hossain
    • 2
  • Mark Thom
    • 2
  1. 1.Queen’s UniversityCanada
  2. 2.University of LethbridgeCanada

Personalised recommendations