Approximating minimum-power degree and connectivity problems

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 with edge costs and degree requirements {r(v):v ∈ V}, the Minimum-Power Edge-Multi-Cover ( ) problem is to find a minimum-power subgraph of so that the degree of every node v is at least r(v). We give an O(logn)-approximation algorithms for , improving the previous ratio O(log⁴ n) of [11]. This is used to derive an O(logn + α)-approximation algorithm for the undirected Minimum-Power k -Connected Subgraph ( ) problem, where is the best known ratio for the min-cost variant of the problem (currently, = O(In k) for n ≥ 2k² and 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 with r(v) ∈ {0,1}, improving over the 2-approximation by [11], and a -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 ( ) problem: find a minimum-power subgraph that contains r(u,v) pairwise edge-disjoint paths for every pair u,v of nodes.