Minimizing the total cost of barrier coverage in a linear domain

Barrier coverage, as one of the most important applications of wireless sensor network (WSNs), is to provide coverage for the boundary of a target region. We study the barrier coverage problem by using a set of n sensors with adjustable coverage radii deployed along a line interval or circle. Our goal is to determine a range assignment \(\mathbf {R}=({r_{1}},{r_{2}}, \ldots , {r_{n}})\) of sensors such that the line interval or circle is fully covered and its total cost \(C(\mathbf {R})=\sum _{i=1}^n {r_{i}}^\alpha \) is minimized. For the line interval case, we formulate the barrier coverage problem of line-based offsets deployment, and present two approximation algorithms to solve it. One is an approximation algorithm of ratio 4 / 3 runs in \(O(n^{2})\) time, while the other is a fully polynomial time approximation scheme (FPTAS) of computational complexity \(O(\frac{n^{2}}{\epsilon })\). For the circle case, we optimally solve it when \(\alpha = 1\) and present a \(2(\frac{\pi }{2})^\alpha \)-approximation algorithm when \(\alpha > 1\). Besides, we propose an integer linear programming (ILP) to minimize the total cost of the barrier coverage problem such that each point of the line interval is covered by at least k sensors.