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 \(\alpha \ge 1\). 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. \(\textsc {SC} _k\) and \(\textsc {OnceSC} _k\) 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 \(\alpha = 1\), and the complexity of the case where \(\alpha >1\) remained open. We extend the above-mentioned NP-hardness result in two ways: We show that OnceSC is NP-hard for every \(\alpha > 1\) and that OnceSC is strongly NP-hard for \(\alpha = 1\). In addition, we show that \(\textsc {OnceSC} _k\), for \(k \ge 2\), is NP-hard, for any \(\alpha \ge 1\), even for uniform radii sets. On the positive side, we present (i) a \(5\gamma ^\alpha \)-approximation algorithm for \(\textsc {OnceSC} _k\), for \(k \ge 1\), where \(\gamma \) is the maximum ratio between two radii associated with the same sensor; (ii) a 5-approximation algorithm for \(\textsc {SC} _k\), for every \(k \ge 1\); and (iii) a \((5+\varepsilon )\)-approximation algorithm for Strip Cover, for any constant \(\varepsilon >0\). Finally, we present an \(O(n \log n)\)-time algorithm for a variant of \(\textsc {OnceSC} _k\) in which all sensors must be activated at the same time.

我们研究了条状覆盖问题 ( SC ) 的变体，其中电池电量有限的传感器部署在线路屏障上，目标是尽可能长时间地覆盖屏障。传感器的能量消耗取决于它的感应半径：能量消耗与传感器半径成正比，提高到常数指数\(\alpha \ge 1\)。在Set Once Strip Cover问题（OnceSC）中，每个传感器的半径可以设置一次，并且可以随时激活传感器。\(\textsc {SC} _k\)和\(\textsc {OnceSC} _k\)是SC和OnceSC的变体，其中每个传感器与一组至多k个预定半径相关联。以前知道，当\(\alpha = 1\)时， OnceSC是 NP 难的，而\(\alpha >1\)保持开放的情况的复杂性。我们以两种方式扩展了上述 NP 难度结果：我们证明了对于每个\(\alpha > 1\) ， OnceSC都是NP-hard，并且对于\(\alpha = 1\)，OnceSC是强NP-hard 。此外，我们证明了\(\textsc {OnceSC} _k\)，对于\(k \ge 2\)，对于任何\(\alpha \ge 1\)都是 NP-hard，即使对于统一的半径集。从积极的方面来说，我们提出了 (i) 一个\(5\gamma ^\alpha \) -对于\(\textsc {OnceSC} _k\)的近似算法，对于\(k \ge 1\)，其中\(\ gamma \)是与同一传感器相关的两个半径之间的最大比率；(ii) 对于每个 \ (\textsc {SC} _k\)的 5 近似算法，对于每个\(k \ge 1\)；(iii) 一个\((5+\varepsilon )\) - Strip Cover的近似算法，对于任何常数\(\varepsilon >0\)。最后，我们针对\(\textsc {OnceSC} _k\)的变体提出了一个\(O(n \log n)\)时间算法其中所有传感器必须同时激活。