Parameterized Algorithms for Power-Efficient Connected Symmetric Wireless Sensor Networks

  • Matthias Bentert
  • René van Bevern
  • André Nichterlein
  • Rolf Niedermeier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10718)


We study an NP-hard problem motivated by energy-efficiently maintaining the connectivity of a symmetric wireless sensor communication network. Given an edge-weighted \(n\)-vertex graph, find a connected spanning subgraph of minimum cost, where the cost is determined by letting each vertex pay the most expensive edge incident to it in the subgraph. We provide an algorithm that works in polynomial time if one can find a set of obligatory edges that yield a spanning subgraph with \(O(\log n)\) connected components. We also provide a linear-time algorithm that reduces any input graph that consists of a tree together with \(g\) additional edges to an equivalent graph with \(O(g)\) vertices. Based on this, we obtain a polynomial-time algorithm for \(g\in O(\log n)\). On the negative side, we show that \(o(\log n)\)-approximating the difference \(d\) between the optimal solution cost and a natural lower bound is NP-hard and that there are presumably no exact algorithms running in \(2^{o(n)}\) time or in \(f(d)\cdot n^{O(1)}\) time for any computable function \(f\).


Monitoring areas and backbones Parameterized complexity Color coding Data reduction Parameterization above lower bounds Approximation hardness Spanning trees 



RvB was supported by the Russian Science Foundation, grant 16-11-10041, while working on Sect. 2. The results in Sects. 3 and 4 were obtained during a research stay of RvB at TU Berlin, jointly supported by TU Berlin, by the Russian Foundation for Basic Research under grant 16-31-60007 mol\(\_\)a\(\_\)dk, and by the Ministry of Science and Education of the Russian Federation under the 5-100 Excellence Programme.


  1. 1.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Althaus, E., Călinescu, G., Mandoiu, I.I., Prasad, S.K., Tchervenski, N., Zelikovsky, A.: Power efficient range assignment for symmetric connectivity in static ad hoc wireless networks. Wirel. Netw. 12(3), 287–299 (2006)CrossRefGoogle Scholar
  3. 3.
    Bentert, M., Fluschnik, T., Nichterlein, A., Niedermeier, R.: Parameterized aspects of triangle enumeration. In: Klasing, R., Zeitoun, M. (eds.) FCT 2017. LNCS, vol. 10472, pp. 96–110. Springer, Heidelberg (2017). CrossRefGoogle Scholar
  4. 4.
    Betzler, N., van Bevern, R., Fellows, M.R., Komusiewicz, C., Niedermeier, R.: Parameterized algorithmics for finding connected motifs in biological networks. IEEE/ACM Trans. Comput. Biol. 8(5), 1296–1308 (2011)Google Scholar
  5. 5.
    Betzler, N., Guo, J., Komusiewicz, C., Niedermeier, R.: Average parameterization and partial kernelization for computing medians. J. Comput. Syst. Sci. 77(4), 774–789 (2011)Google Scholar
  6. 6.
    van Bevern, R., Komusiewicz, C., Sorge, M.: A parameterized approximation algorithm for the mixed and windy capacitated arc routing problem: theory and experiments. Networks (2017, in press)Google Scholar
  7. 7.
    Bruckner, S., Hüffner, F., Karp, R.M., Shamir, R., Sharan, R.: Topology-free querying of protein interaction networks. J. Comput. Biol. 17(3), 237–252 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Carmi, P., Katz, M.J.: Power assignment in radio networks with two power levels. Algorithmica 47(2), 183–201 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Clementi, A.E., Penna, P., Silvestri, R.: On the power assignment problem in radio networks. Mob. Netw. Appl. 9(2), 125–140 (2004)CrossRefzbMATHGoogle Scholar
  10. 10.
    Dost, B., Shlomi, T., Gupta, N., Ruppin, E., Bafna, V., Sharan, R.: Qnet: a tool for querying protein interaction networks. J. Comput. Biol. 15(7), 913–925 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, Heidelberg (2013).
  12. 12.
    Erzin, A.I., Plotnikov, R.V., Shamardin, Y.V.: O nekotorykh polinomial’no razreshimykh sluchayakh i priblizhënnykh algoritmakh dlya zadachi postroyeniya optimal’nogo kommunikatsionnogo dereva. Diskretn. Anal. Issled. Oper. 20(1), 12–27 (2013)Google Scholar
  13. 13.
    Erzin, A.I., Mladenovic, N., Plotnikov, R.V.: Variable neighborhood search variants for min-power symmetric connectivity problem. Comput. Oper. Res. 78, 557–563 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Giacometti, A.: River networks. In: Complex Networks, Encyclopedia of Life Support Systems (EOLSS), pp. 155–180. EOLSS Publishers/UNESCO (2010)Google Scholar
  15. 15.
    Gutin, G., Wahlström, M., Yeo, A.: Rural postman parameterized by the number of components of required edges. J. Comput. Syst. Sci. 83(1), 121–131 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hartung, S., Komusiewicz, C., Nichterlein, A.: Parameterized algorithmics and computational experiments for finding 2-clubs. J. Graph Algorithms Appl. 19(1), 155–190 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hoffmann, S., Wanke, E.: Minimum power range assignment for symmetric connectivity in sensor networks with two power levels (2016). arXiv:1605.01752
  18. 18.
    Impagliazzo, R., Paturi, R.: On the complexity of \(k\)-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mertzios, G.B., Nichterlein, A., Niedermeier, R.: Linear-time algorithm for maximum-cardinality matching on cocomparability graphs. In: MFCS 2017. LIPIcs, vol. 83, pp. 46:1–46:14, Schloss Dagstuhl – Leibniz-Zentrum fuer Informatik (2017)Google Scholar
  21. 21.
    Montemanni, R., Gambardella, L.: Exact algorithms for the minimum power symmetric connectivity problem in wireless networks. Comput. Oper. Res. 32(11), 2891–2904 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Panigrahi, D.: Survivable network design problems in wireless networks. In: Proceedings of 22nd SODA, pp. 1014–1027. SIAM (2011)Google Scholar
  23. 23.
    Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proceedings of 29th STOC, pp. 475–484. ACM (1997)Google Scholar
  24. 24.
    Scott, J., Ideker, T., Karp, R.M., Sharan, R.: Efficient algorithms for detecting signaling pathways in protein interaction networks. J. Comput. Biol. 13(2), 133–144 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sorge, M., van Bevern, R., Niedermeier, R., Weller, M.: From few components to an Eulerian graph by adding arcs. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 307–318. Springer, Heidelberg (2011). CrossRefGoogle Scholar
  26. 26.
    Sorge, M., van Bevern, R., Niedermeier, R., Weller, M.: A new view on rural postman based on Eulerian extension and matching. J. Discrete Alg. 16, 12–33 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Uhlmann, J., Weller, M.: Two-layer planarization parameterized by feedback edge set. Theoret. Comput. Sci. 494, 99–111 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zalyubovskiy, V.V., Erzin, A.I., Astrakov, S.N., Choo, H.: Energy-efficient area coverage by sensors with adjustable ranges. Sensors 9(4), 2446–2460 (2009)CrossRefGoogle Scholar
  29. 29.
    Zhang, H., Hou, J.C.: Maintaining sensing coverage and connectivity in large sensor networks. Ad Hoc Sens. Wirel. Netw. 1(1–2), 89–124 (2005)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Matthias Bentert
    • 1
  • René van Bevern
    • 2
    • 3
  • André Nichterlein
    • 1
  • Rolf Niedermeier
    • 1
  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinBerlinGermany
  2. 2.Novosibirsk State UniversityNovosibirskRussian Federation
  3. 3.Sobolev Institute of MathematicsSiberian Branch of the Russian Academy of SciencesNovosibirskRussian Federation

Personalised recommendations