Near linear time 5/3-approximation algorithms for two-level power assignment problems

We investigate the problem of assigning power levels to nodes of an ad hoc network to minimize total power while preserving connectivity. We consider a simplified version of this problem by requiring bidirected input graphs (ie if an arc uv exists, then the arc vu exists and has the same cost) and that all arcs have cost 0 or 1. This corresponds to a network where each transmitter can operate at high and low power. There are two versions of this problem, a symmetric variant which seeks a connected spanning subgraph and includes an edge in the subgraph if both endpoints have power at least the edge cost, and an asymmetric variant which seeks a strongly connected spanning subgraph and includes an arc in the subgraph if the source endpoint has power at least the arc cost. Both of these have been shown to be NP-Complete. We present 5/3-approximation algorithms for each of these that run in O(mα(n)) where α(n) is the inverse Ackermann function.