Approximating Minimum-Power Degree and Connectivity Problems

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):v∈V}, 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}$ ($\textsf{MP$k$CS}$ ) 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=n−o(n), and is O(log 2 k)=O(log 2 n) for k=n−o(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.