Cross-layer design and operation of wireless networks has attracted significant interest in the last decade, yet some basic problems in the area remain unsolved. In this paper, we consider the joint routing and power control problem, and specifically how to choose transmission powers at the physical layer to maximize stable end-to-end throughput at the network layer for a multi-hop wireless network. This is the back-pressure power control (BPPC) problem. Earlier work had recognized that BPPC is a non-convex problem, and suggested relatively simple suboptimal strategies. Here we show that BPPC is NP-hard. This is a negative result, which however comes with a positive flip side: drawing from related developments in the digital subscriber line (DSL) literature, we devise effective ways to approximate it. We report substantial improvements in transport capacity relative to the earlier state of art, as illustrated in pertinent simulations.