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Power optimization for connectivity problems

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Abstract

Given a graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of the nodes of this graph. Motivated by applications in wireless multi-hop networks, we consider four fundamental problems under the power minimization criteria: the Min-Power b-Edge-Cover problem (MPb-EC) where the goal is to find a min-power subgraph so that the degree of every node v is at least some given integer b(v), the Min-Power k-node Connected Spanning Subgraph problem (MPk-CSS), Min-Power k-edge Connected Spanning Subgraph problem (MPk-CSS), and finally the Min-Power k-Edge-Disjoint Paths problem in directed graphs (MPk-EDP). We give an O(log4 n)-approximation algorithm for MPb-EC. This gives an O(log4 n)-approximation algorithm for MPk-CSS for most values of k, improving the best previously known O(k)-approximation guarantee. In contrast, we obtain an \(O(\sqrt{n})\) approximation algorithm for ECSS, and for its variant in directed graphs (i.e., MPk-EDP), we establish the following inapproximability threshold: MPk-EDP cannot be approximated within O(2log1-ε n) for any fixed ε > 0, unless NP-hard problems can be solved in quasi-polynomial time.

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Authors and Affiliations

  1. Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA

    Mohammad T. Hajiaghayi

  2. Department of Computer Science, Rutgers University-Camden, Camden, NJ, USA

    Guy Kortsarz

  3. Theory of Computation Group, Microsoft Research, Redmond, WA, USA

    Vahab S. Mirrokni

  4. Computer Science Division, The Open University of Israel, Tel-Aviv, Israel

    Zeev Nutov

Authors
  1. Mohammad T. Hajiaghayi
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  2. Guy Kortsarz
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  3. Vahab S. Mirrokni
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  4. Zeev Nutov
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Corresponding author

Correspondence to Mohammad T. Hajiaghayi.

Additional information

This paper was done when V. S. Mirrokni was at Computer Science and Artificial Intelligence Laboratory, MIT.

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Hajiaghayi, M.T., Kortsarz, G., Mirrokni, V.S. et al. Power optimization for connectivity problems. Math. Program. 110, 195–208 (2007). https://doi.org/10.1007/s10107-006-0057-5

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  • DOI: https://doi.org/10.1007/s10107-006-0057-5

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