A note on the minimum power partial cover problem on the plane

Abstract

Given a set of n points and a set of m sensors on the plane, each sensor s can adjust its power p(s) and the covering range which is a disk of radius r(s) satisfying $p(s)=c\cdot r(s)^{\alpha }$. The minimum power partial cover problem, introduced by Freund (Proceedings of international workshop on approximation and online algorithms, pp 137–150. 2011. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.737.1320), is to determine the power assignment on every sensor such that at least k ($k\le n$) points are covered and the total power consumption is minimized. By generalizing the method in Li (Journal of Com. Opti.2020. https://doi.org/10.1007/s10878-020-00567-3) whose approximation ratio is $3^{\alpha }$ and enlarging the radius of each disk in the relaxed independent set, we present an $O(\alpha )$-approximation algorithm.