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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)

Abstract

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 \(f\).

Keywords

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

Notes

Acknowledgments

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.

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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

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