Numerous studies on energy-efficient routing for Multihop Ad Hoc Networks (MANET) look at extending battery life by minimizing the cost at the transmitting node. In this paper, we study the complexity of energy efficient routing when the energy cost of receiving packets is also considered. We first prove that, surprisingly, even when all nodes transmit at the same power, finding a simple unicast path that guarantees enough remaining energy locally at each node in the network then becomes an NP-complete problem. Second, we define formally the problem of finding a virtual backbone that minimized the overall energy cost and prove that this leads to a new NP-complete problem, that we name Connected Exact Cover. Finally, we provide a fully distributed algorithm to reduce the energy drain due to the number of redundant receptions in MANET protocols by offering a modification of the Multi-Point Relay selection scheme and give some provably optimal approximation bounds.