Coordinating Concurrent Transmissions: A Constant-Factor Approximation of Maximum-Weight Independent Set in Local Conflict Graphs

  • Petteri Kaski
  • Aleksi Penttinen
  • Jukka Suomela
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4686)

Abstract

We study the algorithmic problem of coordinating transmissions in a wireless network where radio interference constrains concurrent transmissions on wireless links. We focus on pairwise conflicts between the links; these can be described as a conflict graph. Associated with the conflict graph are two fundamental network coordination tasks: (a) finding a nonconflicting set of links with the maximum total weight, and (b) finding a link schedule with the minimum total length. Our work shows that two assumptions on the geometric structure of conflict graphs suffice to achieve polynomial-time constant-factor approximations: (i) bounded density of devices, and (ii) bounded range of interference. We also show that these assumptions are not sufficient to obtain a polynomial-time approximation scheme for either coordination task.

Keywords

Geometric graphs maximum-weight independent set radio interference 

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References

  1. 1.
    Jain, K., Padhye, J., Padmanabhan, V.N., Qiu, L.: Impact of interference on multi-hop wireless network performance. Wireless Networks 11(4), 471–487 (2005)CrossRefGoogle Scholar
  2. 2.
    Jansen, K.: Approximate strong separation with application in fractional graph coloring and preemptive scheduling. Theoretical Computer Science 302(1–3), 239–256 (2003)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Young, N.E.: Sequential and parallel algorithms for mixed packing and covering. In: Proc. 42nd Annual Symposium on Foundations of Computer Science FOCS, Las Vegas, NV, USA, October 2001, pp. 538–546. IEEE Computer Society Press, Los Alamitos (2001)Google Scholar
  4. 4.
    Håstad, J.: Clique is hard to approximate within n 1 − ε. Acta Mathematica 182, 105–142 (1999)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Khot, S.: Improved inapproximability results for MaxClique, chromatic number and approximate graph coloring. In: Proc. 42nd Annual Symposium on Foundations of Computer Science FOCS, Las Vegas, NV, USA, October 2001, pp. 600–609. IEEE Computer Society Press, Los Alamitos (2001)Google Scholar
  6. 6.
    Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. Journal of the ACM 41(5), 960–981 (1994)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Goldsmith, A.: Wireless Communications. Cambridge University Press, Cambridge, UK (2005)Google Scholar
  8. 8.
    Krishnamachari, B.: Networking Wireless Sensors. Cambridge University Press, Cambridge, UK (2005)Google Scholar
  9. 9.
    Suomela, J.: Approximability of identifying codes and locating-dominating codes. Information Processing Letters 103(1), 28–33 (2007)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Doyle, P.G., Snell, J.L.: Random Walks and Electric Networks. The Mathematical Association of America, Washington, DC, USA (1984)Google Scholar
  11. 11.
    Halldórsson, M.M.: Approximations of independent sets in graphs. In: Jansen, K., Rolim, J.D.P. (eds.) APPROX 1998. LNCS, vol. 1444, pp. 1–13. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  12. 12.
    Halldórsson, M.M.: Approximations of weighted independent set and hereditary subset problems. Journal of Graph Algorithms and Applications 4(1), 1–16 (2000)MathSciNetGoogle Scholar
  13. 13.
    Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time approximation schemes for geometric intersection graphs. SIAM Journal on Computing 34(6), 1302–1323 (2005)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. Journal of the ACM 32(1), 130–136 (1985)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Hunt III, H.B., Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: NC-approximation schemes for NP- and PSPACE-hard problems for geometric graphs. Journal of Algorithms 26(2), 238–274 (1998)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Khanna, S., Motwani, R., Sudan, M., Vazirani, U.: On syntactic versus computational views of approximability. SIAM Journal on Computing 28(1), 164–191 (1999)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. Journal of Computer and System Sciences 43(3), 425–440 (1991)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Chvátal, V., Ebenegger, C.: A note on line digraphs and the directed max-cut problem. Discrete Applied Mathematics 29(2–3), 165–170 (1990)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Erlebach, T., Jansen, K.: Conversion of coloring algorithms into maximum weight independent set algorithms. Discrete Applied Mathematics 148(1), 107–125 (2005)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Petteri Kaski
    • 1
  • Aleksi Penttinen
    • 2
  • Jukka Suomela
    • 1
  1. 1.Helsinki Institute for Information Technology HIIT, Department of Computer Science, University of Helsinki, P.O. Box 68, FI-00014 University of HelsinkiFinland
  2. 2.Networking Laboratory, Helsinki University of Technology, P.O. Box 3000, FI-02015 TKKFinland

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