  Many physical systems, such as water/electricity distribution networks, are
monitored by battery-powered Wireless Sensor Networks (WSNs). Since battery
replacement of sensor nodes is generally difficult, long-term monitoring can be
only achieved if the operation of the WSN nodes contributes to a long WSN
lifetime. Two prominent techniques to long WSN lifetime are i) optimal sensor
activation and ii) efficient data gathering and forwarding based on compressive
sensing. These techniques are feasible only if the activated sensor nodes
establish a connected communication network (connectivity constraint), and
satisfy a compressive sensing decoding constraint (cardinality constraint).
These two constraints make the problem of maximizing network lifetime via
sensor node activation and compressive sensing NP-hard. To overcome this
difficulty, an alternative approach that iteratively solves energy balancing
problems is proposed. However, understanding whether maximizing network
lifetime and energy balancing problems are aligned objectives is a fundamental
open issue. The analysis reveals that the two optimization problems give
different solutions, but the difference between the lifetime achieved by the
energy balancing approach and the maximum lifetime is small when the initial
energy at sensor nodes is significantly larger than the energy consumed for a
single transmission. The lifetime achieved by the energy balancing is
asymptotically optimal, and that the achievable network lifetime is at least
$50$\% of the optimum. Analysis and numerical simulations quantify the
efficiency of the proposed energy balancing approach.
