Maximizing barrier coverage lifetime with static sensors

We study variants of the Strip Cover problem (SC) in which sensors with limited battery power are deployed on a line barrier, and the goal is to cover the barrier as long as possible. The energy consumption of a sensor depends on its sensing radius: energy is drained in inverse proportion to the sensor radius raised to a constant exponent α≥ 1. In the Set Once Strip Cover (OnceSC) the radius of each sensor can be set once, and the sensor can be activated at any time. SC_k and ONCESCk are variants of SC and OnceSC, resp., in which each sensor is associated with a set of at most k predetermined radii. It was previously known that OnceSC is NP-hard when α=1, and the complexity of the case where α> 1 remained open. We extend the above mentioned NP-hardness result in two ways: we show that OnceSC is NP-hard for every α> 1 and that OnceSC is strongly NP-hard for α=1. In addition, we show that ONCESCk, for k≥ 2, is NP-hard, for any α≥ 1, even for uniform radii sets. On the positive side, we present (i) a 5 γ^α-approximation algorithm for ONCESCk, for k≥ 1, where γ is the maximum ratio between two radii associated with the same sensor; (ii) a 5-approximation algorithm for SC_k, for every k≥1; and (iii) a 5 · 2^α-approximation algorithm for Strip Cover. Finally, we present an O(nlog n)-time algorithm for a variant of ONCESCk in which all sensors must be activated at the same time.