Geometric partitioning and robust ad-hoc network design


We present fast approximation algorithms for the problem of dividing a given convex geographic region into smaller sub-regions so as to distribute the workloads of a set of vehicles. Our objective is to partition the region in such a fashion as to ensure that vehicles are capable of communicating with one another under limited communication radii. We consider variations of this problem in which sub-regions are constrained to have equal area or be convex, and as a side consequence, our approach yields a factor 1.99 approximation algorithm for the continuous k-centers problem on a convex polygon.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15


  1. Alzoubi, K. M., Wan, P.-J., & Frieder, O. (2002) Message-optimal connected dominating sets in mobile ad hoc networks. In Proceedings of the 3rd ACM international symposium on mobile ad hoc networking and computing, (pp 157–164).

  2. Aronov, B., Carmi, P., & Katz, M. J. (2009). Minimum-cost load-balancing partitions. Algorithmica, 54(3), 318–336.

    Article  Google Scholar 

  3. Avis, D., & Toussaint, G. T. (1981). An efficient algorithm for decomposing a polygon into star-shaped polygons. Pattern Recognition, 13(6), 395–398.

    Article  Google Scholar 

  4. Bereg, S., Bose, P., & Kirkpatrick, D. (2006). Equitable subdivisions within polygonal regions. Computational Geometry, 34(1), 20–27.

    Article  Google Scholar 

  5. Berman, O., Drezner, Z., Tamir, A., & Wesolowsky, G. O. (2009). Optimal location with equitable loads. Annals of Operations Research, 167(1), 307–325.

    Article  Google Scholar 

  6. Brimberg, J., & Salhi, S. (2005). A continuous location-allocation problem with zone-dependent fixed cost. Annals of Operations Research, 136(1), 99–115.

    Article  Google Scholar 

  7. Carlsson, J. G. (2012). Dividing a territory among several vehicles. INFORMS Journal on Computing, 24(4), 565–577.

    Article  Google Scholar 

  8. Carlsson, J. G., Ge, D., Subramaniam, A., & Ye. Y. (2007). Solving the min–max multi-depot vehicle routing problem. In Proceedings of the FIELDS workshop on global optimization.

  9. Haugland, D., Ho, S. C., & Laporte, G. (2007). Designing delivery districts for the vehicle routing problem with stochastic demands. European Journal of Operational Research, 180(3), 997–1010.

    Article  Google Scholar 

  10. Hochbaum, D. S. (1997). Approximation algorithms for NP-hard problems, volume 20. PWS publishing company Boston.

  11. Melissen, J. B. M., & Schuur, P. C. (2000). Covering a rectangle with six and seven circles. Discrete Applied Mathematics, 99(1), 149–156.

    Article  Google Scholar 

  12. Pavone, M., Arsie, A., Frazzoli, E., & Bullo, F. (2011). Distributed algorithms for environment partitioning in mobile robotic networks. Automatic Control, IEEE Transactions on, 56(8), 1834–1848.

    Article  Google Scholar 

  13. Preparata, F. P., & Shamos, M. I. (1985). Computational geometry: An introduction. New York: Springer.

    Google Scholar 

  14. Royer, E. M., Melliar-Smith, P. M., & Moser, L. E. (2001). An analysis of the optimum node density for ad hoc mobile networks. In IEEE International Conference on Communications, 2001. ICC 2001 (Vol. 3, pp. 857–861).

  15. Stojmenovic, I. (2002). Position-based routing in ad hoc networks. IEEE Communications Magazine, 40(7), 128–134.

    Article  Google Scholar 

  16. Valiente, G. (2013). Algorithms on trees and graphs. Berlin: Springer.

    Google Scholar 

  17. Wattenhofer, R., Li, L., Bahl, P., & Wang, Y.-M. (2001) Distributed topology control for power efficient operation in multihop wireless ad hoc networks. In INFOCOM 2001. Twentieth annual joint conference of the IEEE computer and communications societies. Proceedings. IEEE (Vol. 3, pp. 1388–1397)

  18. Wu J., & Li, H. (1999) On calculating connected dominating set for efficient routing in ad hoc wireless networks. In Proceedings of the 3rd international workshop on Discrete algorithms and methods for mobile computing and communications, (pp. 7–14).

  19. Youla, D. C., & Webb, H. (1982). Image restoration by the method of convex projections: Part 1: Theory. IEEE Transactions on Medical Imaging, 1(2), 81–94.

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to John Gunnar Carlsson.

Additional information

The authors gratefully acknowledge DARPA Young Faculty Award N66001-12-1-4218, NSF Grant CMMI-1234585, and ONR Grant N000141210719.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 197 KB)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Carlsson, J.G., Behroozi, M. & Li, X. Geometric partitioning and robust ad-hoc network design. Ann Oper Res 238, 41–68 (2016).

Download citation


  • Geometric partitioning
  • Network design
  • Location set covering
  • Vehicle routing
  • approximation algorithms