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In this work, we study the problem of minimizing the total power consumption in a multi-hop wireless network subject to a given offered load. It is well-known that the total power consumption of multi-hop wireless networks can be substantially reduced by jointly optimizing power control, link scheduling, and routing. However, the known optimal cross-layer solution to this problem is centralized, and with high computational complexity. In this paper, we develop a low-complexity and distributed algorithm that is provably power-efficient. In particular, under the node exclusive interference model, we can show that the total power consumption of our algorithm is at most twice as large as the power consumption of the optimal (but centralized and complex) algorithm. Our algorithm is not only the first such distributed solution with provable performance bound, but its power-efficiency ratio is also tighter than that of another sub-optimal centralized algorithm in the literature.