Algorithmica

, Volume 60, Issue 4, pp 735–742

# Approximating Minimum-Power Degree and Connectivity Problems

• Guy Kortsarz
• 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 $$\textsf{Minimum-Power k-Connected Subgraph}$$ ($$\textsf{MPkCS}$$ ) 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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## Authors and Affiliations

• Guy Kortsarz
• 1
Email author
• Vahab S. Mirrokni
• 2
• Zeev Nutov
• 3
• Elena Tsanko
• 4
1. 1.Rutgers UniversityCamdenUSA