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 G = (V, E) with edge costs {c_e : e ε E} and degree requirements {r(v) : v ε V}, the Minimum-Power Edge-Multi-Cover (MPEMC) problem is to find a minimum-power subgraph of G so that the degree of every node v is at least r(v). We give an O(log n)-approximation algorithms for MPEMC, improving the previous ratio O(log~4 n) of [11]. This is used to derive an O(log n + α)-approximation algorithm for the undirected Minimum-Power k-Connected Subgraph (MPk-CS) problem, where α is the best known ratio for the min-cost variant of the problem (currently, α = O(ln k) for n ≥ 2k~2 and α = O(ln~2 k · min{n-k, k~(1/2)/log n}) 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(log n)-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 MPEMC with r(v) ∈ {0,1}, improving over the 2-approximation by [11], and a 32/3-approximation for the minimum-power 2-Connected and 2-Edge-Connected Subgraph problems, improving the 4-approximation by [4]. Finally, we give a 4r_(max)-approximation algorithm for the undirected Minimum-Power Steiner Network (MPSN) 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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