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Algorithmica

, Volume 60, Issue 4, pp 735–742 | Cite as

Approximating Minimum-Power Degree and Connectivity Problems

  • Guy KortsarzEmail author
  • Vahab S. Mirrokni
  • Zeev Nutov
  • Elena Tsanko
Article

Abstract

Power optimization is a central issue in wireless network design. Given a graph with costs on the edges, the power of a node is the maximum cost of an edge incident to it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider several fundamental undirected network design problems under the power minimization criteria. Given a graph \(\mathcal{G}=(V,\mathcal{E})\) with edge costs {c(e):e∈ℰ} and degree requirements {r(v):vV}, the \(\textsf{Minimum-Power Edge-Multi-Cover}\) (\(\textsf{MPEMC}\) ) problem is to find a minimum-power subgraph G of \(\mathcal{G}\) so that the degree of every node v in G is at least r(v). We give an O(log n)-approximation algorithms for \(\textsf{MPEMC}\) , improving the previous ratio O(log 4 n). This is used to derive an O(log n+α)-approximation algorithm for the undirected \(\textsf{Minimum-Power $k$-Connected Subgraph}\) (\(\textsf{MP$k$CS}\) ) problem, where α is the best known ratio for the min-cost variant of the problem. Currently, \(\alpha=O(\log k\cdot \log\frac{n}{n-k})\) which is O(log k) unless k=no(n), and is O(log 2 k)=O(log 2 n) for k=no(n). Our result shows that the min-power and the min-cost versions of the \(\textsf{$k$-Connected Subgraph}\) problem are equivalent with respect to approximation, unless the min-cost variant admits an o(log n)-approximation, which seems to be out of reach at the moment.

Keywords

Power Graphs Wireless Degree k-connectivity Approximation 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Guy Kortsarz
    • 1
    Email author
  • Vahab S. Mirrokni
    • 2
  • Zeev Nutov
    • 3
  • Elena Tsanko
    • 4
  1. 1.Rutgers UniversityCamdenUSA
  2. 2.Google ResearchNew YorkUSA
  3. 3.The Open University of IsraelRaananaIsrael
  4. 4.IBMHaifaIsrael

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