On the Hardness of Range Assignment Problems

  • Bernhard Fuchs
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3998)


We investigate the computational hardness of the Connectivity, the Strong Connectivity and the Broadcast type of Range Assignment Problems in ℝ2 and ℝ3. We present new reductions for the Connectivity problem, which are easily adapted to suit the other two problems. All reductions are considerably simpler than the technically quite involved ones used in earlier works on these problems. Using our constructions, we can for the first time prove NP-hardness of these problems for all real distance-power gradients α > 0 (resp. α > 1 for Broadcast) in 2-d, and prove APX-hardness of all three problems in 3-d for allα > 1. Our reductions yield improved lower bounds on the approximation ratios for all problems where APX-hardness was known before already. In particular, we derive the overall first APX-hardness proof for Broadcast. This was an open problem posed in earlier work in this area, as was the question whether (Strong) Connectivity remains NP-hard for α = 1. Additionally, we give the first hardness results for so-called well-spread instances.


Planar Graph Vertex Cover Strong Connectivity Vertex Cover Problem Minimal Vertex Cover 
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  1. 1.
    Baker, B.: Approximation Algorithms for NP-Complete Problems on Planar Graphs. Journal of the ACM 41(1), 153–180 (1994)MATHCrossRefGoogle Scholar
  2. 2.
    Berman, P., Karpinski, M.: On Some Tighter Inapproximability Results. In: Proc. 26th ICALP, pp. 200–209 (1999), Also available as ECCC Report TR98-065 at http://eccc.uni-trier.de/eccc/
  3. 3.
    Călinescu, G., Măndoiu, I., Zelikovsky, A.: Symmetric Connectivity with Minimum Power Consumption in Radio Networks. In: Proc. IFIP TCS, vol. 223, pp. 119–130 (2002)Google Scholar
  4. 4.
    Chlebík, M., Chlebíková, J.: Inapproximability results for bounded variants of optimization problems. In: Lingas, A., Nilsson, B.J. (eds.) FCT 2003. LNCS, vol. 2751, pp. 27–38. Springer, Heidelberg (2003), Also available as ECCC Report TR03-026Google Scholar
  5. 5.
    Clementi, A., Crescenzi, P., Penna, P., Rossi, G., Vocca, P.: A Worst-Case Analysis of an MST-based Heuristic to Construct Energy-Efficient Broadcast Trees in Wireless Networks. Technical Report 010, University of Rome “Tor Vergata”, Math. Department (2001)Google Scholar
  6. 6.
    Clementi, A., Huiban, G., Penna, P., Rossi, G., Verhoeven, Y.: Some Recent Theoretical Advances and Open Questions on Energy Consumption in Ad-Hoc Wireless Networks. In: Proc. 3rd Workshop on Approximation and Randomization Algorithms in Communication Networks (ARACNE), pp. 23–38 (2002)Google Scholar
  7. 7.
    Clementi, A., Penna, P., Silvestri, R.: On the Power Assignment Problem in Radio Networks. Mobile Networks and Applications 9(2), 125–140 (2004), Also available as ECCC Report TR00-054Google Scholar
  8. 8.
    Eades, P., Stirk, C., Whitesides, S.: The techniques of Kolmogorov and Bardzin for three-dimensional orthogonal graph drawings. Information Processing Letters 60(2), 97–103 (1996)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Garey, M., Johnson, D.: The Rectilinear Steiner Tree Problem is NP-Complete. SIAM Journal of Applied Mathematics 32(4), 826–834 (1977)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Garey, M., Johnson, D., Stockmeyer, L.: Some simplified NP-Complete Problems. Theoretical Computer Science 1, 237–267 (1976)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Karp, R.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds.) Complexity of Computer Computations, pp. 85–103 (1972)Google Scholar
  12. 12.
    Kirousis, L., Kranakis, E., Krizanc, D., Pelc, A.: Power consumption in packet radio networks. Theoretical Computer Science 243, 289–305 (2000)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Pahlavan, K., Levesque, A.: Wireless Information Networks. Wiley-Interscience, Hoboken (1995)Google Scholar
  14. 14.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. Journal of Computer and System Sciences 43, 425–440 (1991)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Rossi, G.: The Range Assignment Problem in Static Ad-Hoc Wireless Networks (PhD-thesis), University of Siena (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Bernhard Fuchs
    • 1
  1. 1.Zentrum für Angewandte Informatik KölnUniversität zu KölnKölnGermany

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