Maximizing Barrier Coverage Lifetime with Static Sensors

Abstract

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 $\alpha \ge 1$. 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. $\textsc {SC} _k$ and $\textsc {OnceSC} _k$ 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 $\alpha = 1$, 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 $5 \cdot 2^\alpha$-approximation algorithm for Strip Cover. 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.

D. Rawitz—Supported by the Israel Science Foundation (grant no. 497/14).