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 OnceSC _k 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 OnceSC _k , for k ≥ 2 , is NP-hard, for any α≥ 1 , even for uniform radii sets. On the positive side, we present (i) a 5γ ^α -approximation algorithm for OnceSC _k , 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(n log n) -time algorithm for a variant of OnceSC _k in which all sensors must be activated at the same time.