The minimum range assignment problem consists of assigning transmission ranges to the stations of a multi-hop packet radio network so as to minimize the total power consumption provided that the transmission range assigned to the stations ensures the strong connectivity of the network (i.e. each station can communicate with any other station by multi-hop transmission). The complexity of this optimization problem was studied by Kirousis, Kranakis, Krizanc, and Pelc (1997). In particular, they proved that, when the stations are located in a 3-dimensional Euclidean space, the problem is NP-hard and admits a 2-approximation algorithm. On the other hand, they left the complexity of the 2-dimensional case as an open problem.

As for the 3-dimensional case, we strengthen their negative result by showing that the minimum range assignment problem is APX-complete, so, it does not admit a polynomial-time approximation scheme unless P=NP.

We also solve the open problem discussed by Kirousis et al. by proving that the 2-dimensional case remains NP-hard.


Transmission Range Vertex Cover Hardness Result Communication Graph Multihop Radio Network 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alimonti, P., Kann, V.: Hardness of approximating problems on cubic graphs. In: Bongiovanni, G., Bovet, D.P., Di Battista, G. (eds.) CIAC 1997. LNCS, vol. 1203, pp. 288–298. Springer, Heidelberg (1997)Google Scholar
  2. 2.
    Arikan, E.: Some complexity results about packet radio networks. IEEE Transactions on Information Theory IT-30, 456–461 (1984)MathSciNetGoogle Scholar
  3. 3.
    Baker, B.S.: Approximation algorithms for np-complete problems on planar graphs. Journal of ACM 41, 153–180 (1994)zbMATHCrossRefGoogle Scholar
  4. 4.
    Berman, P., Karpinski, M.: On some tighter inapproximability results. Electronic Colloquium on Computational Complexity 29 (1998)Google Scholar
  5. 5.
    Eades, P., Symvonis, A., Whitesides, S.: Two algorithms for three dimensional orthogonal graph drawing. In: Graph Drawing 1996. LNCS, vol. 1190, pp. 139–154 (1996)Google Scholar
  6. 6.
    Ephemides, A., Truong, T.: Scheduling broadcast in multihop radio networks. IEEE Transactions on Communications 30, 456–461 (1990)CrossRefGoogle Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability - A Guide to the Theory of NP-Completness. Freeman and Co., New York (1979)Google Scholar
  8. 8.
    Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica. Special Issue on Graph Drawing 16, 4–32 (1996) (Extended Abstract in 33-th IEEE FOCS (1992))zbMATHMathSciNetGoogle Scholar
  9. 9.
    Kirousis, L.M., Kranakis, E., Krizanc, D., Pelc, A.: Power consumption in packet radio networks. In: Reischuk, R., Morvan, M. (eds.) STACS 1997. LNCS, vol. 1200. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  10. 10.
    Pahlavan, K., Levesque, A.: Wireless Information Networks. Wiley-Interscince, New York (1995)Google Scholar
  11. 11.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and com- plexity classes. J. Comput. System Science 43, 425–440 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley Publishing Company, Inc., Reading (1994)zbMATHGoogle Scholar
  13. 13.
    Ramanathan, S., Lloyd, E.: Scheduling boradcasts in multi-hop radio networks. IEEE/ACM Transactions on Networking 1, 166–172 (1993)CrossRefGoogle Scholar
  14. 14.
    Ramaswami, R., Parhi, K.: Distributed scheduling of broadcasts in radio network. in: INFOCOM, pp. 497–504 (1989)Google Scholar
  15. 15.
    Valiant, L.: Universality considerations in vlsi circuits. IEEE Transactions on Computers C-30, 135–140 (1981)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Andrea E. F. Clementi
    • 1
  • Paolo Penna
    • 1
  • Riccardo Silvestri
    • 2
  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata” 
  2. 2.Dipartimento di Matematica Pura e ApplicataUniversit‘a de L’Aquila 

Personalised recommendations