1.61-approximation for min-power strong connectivity with two power levels

Abstract

Given a directed simple graph $G=(V,E)$ and a cost function $c:E \rightarrow R_+$, the power of a vertex $u$ in a directed spanning subgraph $H$ is given by $p_H(u) = \max _{uv \in E(H)} c(uv)$, and corresponds to the energy consumption required for wireless node $u$ to transmit to all nodes $v$ with $uv \in E(H)$. The power of $H$ is given by $p(H) = \sum _{u \in V} p_H(u)$. Power Assignment seeks to minimize $p(H)$ while $H$ satisfies some connectivity constraint. In this paper, we assume $E$ is bidirected (for every directed edge $e \in E$, the opposite edge exists and has the same cost), while $H$ is required to be strongly connected. Moreover, we assume $c:E \rightarrow \{A,B\}$, where $0 \le A < B$. We improve the best known approximation ratio from 1.75 (Chen et al. IEEE GLOBECOM 2005) to $\pi ^2/6 - 1/36 + \epsilon \le 1.61$ using an adaptation of the algorithm developed by Khuller et al. [SIAM J Comput 24(4):859–872 1995, Discr Appl Math 69(3):281–289 1996] for (unweighted) Minimum Strongly Connected Subgraph.