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 proportion to the sensor radius raised to a constant exponent α≥ 1. In the Set Once Strip Cover problem (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, respectively, 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 + ε) -approximation algorithm for Strip Cover, for any constant ε> 0. 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.