LATIN 2008: LATIN 2008: Theoretical Informatics pp 423-435

# Approximating Minimum-Power Degree and Connectivity Problems

• Guy Kortsarz
• Vahab S. Mirrokni
• Zeev Nutov
• Elena Tsanko
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4957)

## Abstract

Power optimization is a central issue in wireless network design. Given a (possibly directed) graph with costs on the edges, the power of a node is the maximum cost of an edge leaving 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 Open image in new window with edge costs Open image in new window and degree requirements {r(v):v ∈ V}, the Minimum-Power Edge-Multi-Cover (Open image in new window) problem is to find a minimum-power subgraph of Open image in new window so that the degree of every node v is at least r(v). We give an O(logn)-approximation algorithms for Open image in new window, improving the previous ratio O(log4n) of [11]. This is used to derive an O(logn + α)-approximation algorithm for the undirected Minimum-Power k-Connected Subgraph (Open image in new window) problem, where Open image in new window is the best known ratio for the min-cost variant of the problem (currently, Open image in new window for n ≥ 2k2 and Open image in new window otherwise). Surprisingly, it shows that the min-power and the min-cost versions of the k-Connected Subgraph problem are equivalent with respect to approximation, unless the min-cost variant admits an o(logn)-approximation, which seems to be out of reach at the moment. We also improve the best known approximation ratios for small requirements. Specifically, we give a 3/2-approximation algorithm for Open image in new window with r(v) ∈ {0,1}, improving over the 2-approximation by [11], and a $$3\frac{2}{3}$$-approximation for the minimum-power 2-Connected and 2-Edge-Connected Subgraph problems, improving the 4-approximation by [4]. Finally, we give a 4 r max -approximation algorithm for the undirected Minimum-Power Steiner Network (Open image in new window) problem: find a minimum-power subgraph that contains r(u,v) pairwise edge-disjoint paths for every pair u,v of nodes.

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© Springer-Verlag Berlin Heidelberg 2008

## Authors and Affiliations

• Guy Kortsarz
• 1
• Vahab S. Mirrokni
• 2
• Zeev Nutov
• 3
• Elena Tsanko
• 4
1. 1.Rutgers University, Camden
2. 2.Microsoft Research
3. 3.The Open University of Israel, Raanana
4. 4.IBM, Haifa